Library 


UNIVERSITY   OF   CALIFORNIA, 

DEPARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY,  CALIFORNIA 


.J&  .W 


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ESTABLISHED    1845 

W.  &  L.  E.  GURLEY,  TROY,  N.  Y. 

MANUFACTURERS  OF 

Civil  Engineers'  and  Surveyors'  Instruments 


All  Instruments  Sent  to  the  Purchaser  Adjusted 
and  Ready  for  Use.     Send  for  Full  Illus- 
trated Price  List  and  Circular. 


A  MANUAL 


LAND  SURVEYING 


COMPRISING 


AN  ELEMENTARY  COURSE  OF  PRACTICE 
WITH  INSTRUMENTS 


AND  A  TREATISE  UPON  THE 


Survey  of  Public  and  Private  Lands, 


PREPARED 


For  use  of  Schools  and  Surveyors. 


BY  F.  HODGMAN,  M.  S.,  C.  E., 

Practical  Surveyor  and  Engineer. 


'Let  things  that  have  to  be  done  be  learned  by  doing  them." 


THE  F.   HODGMAN   CO., 

CLIMAX,    MICHIGAN. 
1907. 


TA55/ 


Engineering 

Library 
Entered  according  to  Act  of  Congress  in  the  year  1903. 

BY  F.  HODGMAN, 
In  the  office  of  the  Librarian  of  Congress,  at  Washington 


PREFACE. 


This  addition  to  the  already  numerous  treatises  on  land 
surveymg  was  caused  by  the  demand  of  the  surveyors  of 
Michigan  for  a  treatise  which  would  deal  with  the  prac- 
tical questions  which  meet  the  surveyor  in  his  every 
day  work  in  the  field.  Several  admirable  treatises  were 
already  in  existence  which  dealt  amply  with  the  mathe- 
matical and  instrumental  part  of  surveymg.  But  the 
perplexing  questions  which  meet  the  surveyor  are  not 
questions  of  mathematical  calculation  or  of  the  use  of 
instruments.  On  the  contrary  they  are,  for  the  most  part, 
questions  of  how  to  apply  the  principles  of  common  law 
and  statutory  enactment  to  the  location  of  boundary 
lines.  These  are  the  controlling  considerations  in  all  re- 
surveys;  a  class  which  comprises  probably  nine-tenths  of 
all  the  land  surveys  which  are  made.  Scarcely  an  allusion 
to  these  principles  was  to  be  found  in  any  of  the  works 
on  surveying  extant.  In  1880  the  Michigan  Association 
of  Surveyors  and  Civil  Engineers  appointed  a  committee 
on  manual,  to  prepare  a  work  which  would  give  authori- 
tative answers  to  the  many  questions  of  practice  which 
came  up  before  them.  The  committee  spent  their  spare 
time  for  five  years  in  an  exhaustive  research  of  the  laws 
and  the  decisions  of  the  highest  courts  in  the  land.  The 
chairman  attended  the  meetings  of  various  surveyors' 
associations  and  collected  their  reports.  From  the  great 
mass  of  material  thus  collected,  the  leading  points  in  the 
laws  of  the  United  States  and  the  decisions  of  the  courts 
of  last  resort  were  selected,  covering,  as  nearly  as  possible, 
all  the  points  relative  to  surveys  and  boundary  lines 
which  arise  in  the  land  surveyor's  practice  The  legal 
decisions  quoted  are  a  part  of  the  Common  iav?  ol  the 
whole  country  and  apply  wherever  the  Common  law 
prevails,  whetf-d  England,  or  the  United 


IV  PREFACE. 

States.  It  should  be  remembered,  however,  that  differ- 
ent courts  do  not  always  expound  the  law  alike,  and 
sometimes  a  court  reverses  its  own  decisions.  When- 
ever there  appears  to  be  a  conflict  of  authorities,  the 
Surveyor  should  follow  the  latest  decisions  in  his  own 
State  if  there  be  any.  It  seemed  to  the  committee 
to  be  important  that  the  student  in  land  surveying 
should  be  taught  these  things ;  that  they  were  as 
necessary  for  the  beginner  to  know  as  for  the  older 
practitioner,  and  hence  might  properly  be  incorporated 
in  the  text  book.  Having  this  in  view,  it  was  decided  to 
extend  the  scope  of  the  manual  by  including  such  mathe- 
matical work  as  would  make  it  equally  adapted  to  the 
use  of  the  student  as  a  text  book  and  the  practical  sur- 
veyor as  a  book  of  reference.  In  preparing  this  portion 
of  the  work,  the  leading  idea  has  been  that,  so  far  as 
possible,  the  student  should  be  taught  by  actual  practice 
in  the  field,  as  well  as  in  the  class  room;  that  he  should 
learn  to  survey  by  surveying.  The  solution  of  a  problem 
in  surveying  in  actual  practice  is  always  worked  out 
upon  the  ground,  hence  suggestions  are  made  to  the 
student  how  problems  may  be  solved,  instead  of  giving 
any  formal  solution.  It  is  pre-supposed  that  every 
successful  teacher  will  have  methods  of  his  own  for 
conveying  instruction,  and  will  use  these  suggestions  or 
make  different  ones  as  may  seem  best  to  him.  Doubtless 
things  have  been  omitted  which  some  would  regard 
as  important  to  have  introduced.  Such  omissions  will  be 
supplied  by  teachers  at  their  pleasure  and  convenience. 
We  acknowledge  our  indebtedness  to  the  authors  of 
many  treatises  which  have  been  consulted  in  the  prepara- 
tion of  this  volume,  especially  to  the  works  of  Davies, 
Oillespie,  Hawes  and  Dunn,  also  to  Messrs.  W.  &  L.  E. 
Gurley  for  many  favors  received,  and  to  the  officers  and 
members  of  the  Surveyors'  Associations  of  Michigan, 
Ohio,  Indiana,  Illinois  and  Missouri  for  many  valuable 
suggestions,  sympathy  and  assistance. 

F.  HODGMAN. 
Climax,  Mich., 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

DEFINITIONS. 

PAGE. 

.INSTRUMENTS  FOR  MEASURING  DISTANCES. 

The  Chain 2 

The  Steel  Tape 3 

Marking  Pins 4 

Measuring 4 

MEASURES  OF  LENGTH  AND  AREA,  English 9 

Old  Spanish 11 

Old  French 13 

Standard  Measures 14 

THE  PICKET,  To  Run  Line  with 16 

To  Pass  Obstacles- 18 


CHAPTER  II. 

INSTRUMENTS. 

THE  SURVEYOR'S  COMPASS,  Description  of 19 

Adjustments  of 22 

Electricity 24 

To  Run  Lines  with 24 

To  Pass  Obstacles 25 

THE  MAGNETIC  NEEDLE,  Changes  in  direction  of 26 

Local  Attraction 27 

Difference  in  Instruments 27 

Things  to  be  Observed ___  27 

Marking  Lines — 28 

How  to  find  a  True  Meridian 28 

THE  TRANSIT,  Description  and  Adjustments 52 

How  to  use — - — •— -- 59 

Assistants  and  their  Duties 60 

The  Color  Pole 61 

Projecting  the  Line 62 

[v] 


VI  TABLE  OF  CONTENTS. 

CHAPTER  III. 

INSTRUMENTS,  CONTINUED. 

THE  SOLAR  COMPASS,  Description  and  Adjustments 65 

How  to  use - 73 

SOLAR  ATTACHMENT  TO  TRANSIT. 

Description  and  Adjustments 80 

How  to  use 86 

CHAPTER  IV. 

MEASUREMENT  OF  ANGLES. 

TO  MEASURE  ANGLES,  With  Tape  and  Tins 88 

With  the  Compass — 91 

With  the  Transit 100 

Verniers 102 

TO  CORRECT  RANDOM  LINES,  Of  one  course 93 

Of  several  courses 96 

CHAPTER  V. 

PASSING    OBSTACLES    AND    MEASURING    INACCESSIBLE 
DISTANCES. 

PASSING  OBSTACLES,  By  Parallel  Lines 107 

By  60°  Angles —  107 

TO  MEASURE  INACCESSIBLE  DISTANCES,  By  triangles.-  10S 

Stadia  Measures 110 

The  Gradienter 115 

CHAPTER  VI. 

PLATTING  AND  COMPUTING  AREAS. 

PLATTING,  Instruments  used 119 

COMPUTING  AREAS,  Triangles 122 

Quadrangles :  Rectangles,  Trapezoids  and  Trapeziums—  124 

Irregular  Polygons 125 

Offsets 120 

Rectangular  Coordinates 128 

Application  to  Area 129 

The  Traverse  Table 134 

Meridian  Distances 136 

Supplying  Omissions 143 

Reducing  Irregular  Polygons 148 

Division  and  Partition  of  Land ^-  153 

Method  by  Approximations 163 

Field  Notes 164 

Abridging  Field  Notes 168 


TABLE  OF  CONTENTS.  TIT 

CHAPTER  VII. 

CURYELINEAR  SURVEYING. 

Preliminary  Propositions 170 

To  run  Curves  with  Picket  and  Tape 171 

Field  Notes  of  Transit  Lanes 173 

To  run  a  Curve  with  the  Transit,  Different  Methods 171 

To  locate  a  Curve  from  its  Middle  Point 178 

To  locate  a  Curve  from  some  Intermediate  Point 179 

To  locate  a  Curve  from  Point  of  Intersection jgO 

Passing  Obstructions  in  Line  of  Curve 181 

Compound  Curves 1&J 

Useful  Formula 184 


CHAPTER  VIII. 

ORIGINAL    SURVEYS. 

Surveys,  Classified 186 

Original  Surveys,  Government  and  Private 186 

PUBLIC  DOMAIN,  How  and  when  Acquired 1S8 

Amount  of - — — - 189 

Origin  of  Systems  of  Surveys  of 189 

Laws  relating  to  Survey  of,  where  found 193 

U.  S.  LAWS  RELATING  TO  SURVEYS  OF  PUBLIC  LANDS. 

Appointment  of  Surveyor  General 193 

Qualifications  of 1H3 

Term  of  Office 194 

When  Records  and  Field  Notes  to  be  turned  over  to 

the  State 194 

Discontinuance  of  Office 194 

When  Authority  to  vest  in  Com.  of  Gen.  Land  Office 193 

Free  Access  to  Field  Notes  and  Records If5 

Surveyor  General  to  Employ  Deputies 183 

To  cause  Survey  of  Base  and  Meridian  Lines 196 

To  cause  Survey  of  Private  Land  Claims .  177-195 

To  inspect  Surveys  in  Person  or  by  Agent 197 

Pay  of  Agent 197 

Deputy  Surveyor  to  Give  Bond 197 

Deputy  Surveyor  to  make  Oath  to  Field  Notes 198 

Penalty  for  Fraudulent  Survey 198 

PUBLIC  LANDS,  How  Divided  into  Townships 198 

Township  Lines,  how  marked 199 

Townships,  how  subdivided  into  Sections 199 

Sections,  how  numbered 171-199 

Section  Corners,  how  marked 199 

Excess  or  Deficiency  over  six  miles 199 

Lines,  how  marked  and  measured 200 


VII!  TABLE  OF  CONTENTS. 

what  Surveyors  to  note  in  Field  Books 300 

Disposition  of  Field  Books  and  making  of  Plats, ...  200 

SECTIONS  AND  SUBDIVISIONS  OF  SECTIONS, 

How  Boundaries  and  Contents  are  found ,  200 

U.  8,  Survey  Corners  the  true  ones '. ,  ,  , 201 

Corners  of  X  and  ^  Sections  not  set  by  Government 

Survey. . 201 

Boundary  Linee  of  U.  8.  Survey  the  true  ones. ....  201 

Those  not  run,  how  found 201 

True  Contents  of  Sections  returned 201 

True  Contents  of  %  and  ^  Sections  which  are  not 

Returned * ,  201 

Fractional  Sections,  how  divided 202 

When  ordinary  Course  may  be  departed  from. .....  202 

Surveys  in  Nevada,  Oregon,  and  California 203 

When  Rectangular  System  may  be  departed  from . .  203 

Instructions,  a  part  of  Contract 20i 

Survey  of  Mining  Claims  and  Rights  of  Owners. ...  201 

Appointment  of  Mineral  Surveyors 206 

Plats  and  Field  Notes  of  Mining  Surveys .  ,  207 

Contracts  to  be  approved  by  Com.  Gen.  Land  Office  207 

Commissioner  to  fix  Prices  for  Surveys,  etc 207 

Extra  Price  in  Oregon,  Washington,  and  California  208 

Penalty  for  Interference  with  Surveys 209 

Surveyors  appointed  to  select  Timber  Lands 210 

Duty  of  Director  of  Geological  Survey 210 

FIELD  WORK  AND  CHANGES  THAT  HAVE  BEEN 
MADE. 

Two  Mile  Blocks,  Act  of  1796  211 

Subdivisions  into  half  Sections,  Act  of  1800 211 

Changes  in  manner  of  Subdividing,  Double  Corners, 

etc , .  211 

How  Area  of  Fractions  is  Calculated 213 

INSTRUCTIONS  OF  1902. 

System  of  rectangular  Surveying 218 

Establishment  of  Meridians,  Base  Lines,  and  Parallels  219 

Division  into  Townships  and  Sections. 220 

Excess  or  deficiency  in  measurement 220 

How  Townships  and  Sections  are  numbered , . .  220 

Instruments  to  be  used 221 

Tests  and  Adjustments  of ,  .  , . .  221 

Chains  and  Tally  pins .     222 

Process  of  chaining 223 

Leveling  chain  and  Plumbing  phio 223 

Marking  lines 224 

Marking  random  lines 225 

Insuperable  objects  in  line 226 

Witness  Points  where  made. 226 

Establishing  Corners '     2£6 

Marking  Tools 227 

Surveying  Monuments 227 

Descriptions  of  Corners 227 


TABLE   OF  CONTENTS.  IX 

Abbreviations 228 

Standard  Township  Corners,  how  marked 230 

Witness  Corners,  how  marked 233 

Witness  Corners  in  Roads 233 

Witness  points,  how  marked 234 

Corners  on  rock 234 

Location  of  Mounds 234 

Mounds  of  Stone '234 

Bearing  Trees 235 

Stones  for  Corners 237 

When  Lines  to  be  discontinued  at  Corners 237 

Marks  to  be  Cut    237 

Orientation  of  Corners 237 

Size  of  Posts,  Mounds,  etc 

Corner  Materials 237 

Initial  Points ., 238 

Base  Line 238 

Principal  Meridians 240 

Standard  Parallels 040 

Guide  Meridians 241 

Township  Exteriors 241 

Township  exteriors  where  impassable  objects  occur.  o4o 

Method  of  Subdividing 243 

Method  of  Subdividing,  Exceptions 247 

Meandering  Streams 249 

Meandering  Lakes 251 

Objects  to  be  noted 253 

Prescribed  Limits  for  Closings  and  Lengths  of  Lines  255 

Field  Notes  Blank  Books  furnished 256 

What  Original  Field  Notea  are 256 

SURVEYING  BASE  LINES  AND  STANDARD  PAR 
ALLELS  BY  OFFSETS  FROM  STRAIGHT 
LINES. 

Secant  Method  and  Tables 257 

Tangent  Method  and  Tables 264 

- 


CHAPTER  IX. 

SUBDIVISION   OF  SECTIONS. 

SUBDIVISION  OF  SECTIONS 267 

Four  Different  Cases 269 

Quarter  Sections 270 

Half-Quarter  Sections 270 

Fractional  Sections 270 

.... 


X  TABLE  OF  CONTENTS. 

Section  Six 271 

Sections  made  Fractional  by  Waters 272 

Irregular  Subdivision  of  Sections  made  Fractional  by  Waters—  273 

Exceptional  Cases 274 

Sample  Resurvey  and  Subdivision  of  a  Section 274 

Private  Surveys 283 

Higbway  Surveys 284 

Surveys  for  Town  Plats 284 

What  Plats  in  Michigan  must  contain 285 

What  Record  of  Plats  in  Michigan  must  contain 285 

Monuments ...  287 


CHAPTER  X. 

RESURVEYS. 

RESUEVEYS. 

Authority  of  Surveyor 2^'J 

What  the  Surveyor  is  called  on  to  do 289. 

DECISIONS  OF  SUPREME  COURTS,  Giving— 

Rules  for  construing  Descriptions  of  Land 289 

Adverse  Possession 305 

Rules  of  Construction  when  Land  borders  on  Waters—  303 

How  to  Locate  Corners  and  Boundary  Lines 317 

General  Rules 317 

Alluvium ., 337 

Rules  Applicable  to  U.  S.  Survey. 339 

Mineral  Surveys 350 

How  to  Write  Descriptions  for  Deeds 351 


CHAPTER  XI. 

RE-LOCATING  LOST  CORNERS. 

General  Rule 358 

Lost  Corners  of  U.  S.  Survey  in  Base  Lines,  etc 357 

Lost  Closing  Section  Corners 357 

Lost  Interior  Section  Corners 358 

Lost  Township  Corners . 358 

Lost  Quarter-Section  Corners 353 

Lost  Meander  Corners. '. 353 

Exceptional  Methods 359 

HOW  TO  FIND  LOST  CORNERS,  Evidences  of 

Original  Posts — 360 

Bearing  Trees .___ , 361 

Fences — _ 362 

Distant  Corners 363 

Persons 364 


TABLE  OF  CONTENTS.  XI 

CHAPTER  XII. 

MISCELLANEOUS. 

Miscellaneous  Questions 365 

RIGHTS,  DUTIES,  ETC.,  OF  SURVEYORS 377 

To  fix  Lines  by  Consent  of  Parties 377 

Have  no  Authority  of  their  own  for  that  purpose 377 

Or  to  determine  where  Corners  and  Lines  are—  377 

Old  Boundaries  not  to  be  disturbed 378 

County  Surveyor's  Certificate  not  Admissible  in  evi- 
dence in  Michigan 378 

Surveyor  Liable  for  Damages  for  Unskillful  Work 378 

Judicial  Functions  of  Surveyors 379 


CHAPTER 

LEVELING   AND   DRAINAGE  SURVEYING. 

Definitions 895 

Difference  between  True  and  Apparent  Level   395 

Instruments  for  Leveling 390 

The  Wye  Level  and  its  Adjustments 397 

Leveling  Rods,  Target  and  Speaking 401 

To  find  Difference  in  Level  of  Different  Points 403 

Drawing  Profile 408 

Drainage  Surveying 409 


TABLES. 

Suggestions  to  Young  Surveyors i-iv 

Trigonometrical  Fonnulae I rv-vi 

Table  of  Logarithms 1-16 

Natural  Sines  and  Cosines 18-  26 

Natural  Tangents 28-  39 

Logarithmic  Sines  and  Tangents 40-84 

Traverse  Table 86-  91 

Departures _.  92 

Natural  Secants 93-  94 

Azimuth  of  Polaris  at  Elongation 94 

Gradienter  Tables 95 

Mean  Refractions 96 

Acreage  of  Open  Drains 97 

Acreage  of  Tile  Drains  and  Capacity  of  Tile 98 


XII  TABLE  OP   CONTENTS. 

Azimuths  of  Tangent 

Offsets  from  Tangent 100 

Minutes  in  Decimals  of  a  Degree 101 

Inches  in  Decimals  of  a  Foot 101 

Radii  and  Deflections 101 

Tangents  and  Externals  of  a  1°  Curve 102-105 

Curve  Formulae 113 

Stadia  Reductions  for  Reading  100 106-112 


OF 


LAND  SURVEYING. 


CHAPTEE  I. 
I.   DEFINITIONS.    FIELD  WORK,  &c. 

1.  Land  Surveying  is  the  art  of  measuring  distances 
and  running  lines  on  the  earth's  surface  to  determine 
the  boundaries  or  to  ascertain  the  areas  of  tracts  of  land. 
The  lines  run  are  not  mathematical  lines,  but  are  repre- 
sentations of  them,  traced  upon  the  earth's  surface  by 
means  of  various  instruments,  and  marked  to  the  eye 
by  chops  and  notches  cut  upon  trees,  or  rocks,  or  by  stakes 
or  stones  set  in  the  ground,  or  any  other  means  to  render 
them  visible. 

2.  Original  Surveys  are  the  surveys  which  are 
first  made  for  the  purpose  of  locating  upon  the  ground 
the  boundaries  of  tracts  of  land,  and  marking  them  by 
visible   objects.    This   work  is  called  the  Field  Work. 
A  full  description  of  what  is  done  is  kept  by  the  sur- 
veyor and  is   called   the   field   notes.    The   field   notes 
furnish  the  data  from  which    to  make  a  map  of  the 
land    and    calculate   the   area.    They   also  furnish  the 
evidence  from  which  to  again   find   and   identify   the 
boundaries  upon  the  ground. 

3.  Resurveys  are  those  which  are  made  for  the  pur- 
pose  of   finding   the   boundaries    which    were  marked 
when  the  original  survey  was  made. 


2  A  MANUAL   OF   LAND    SURVEYING. 

4.  The  instruments  most  commonly  used  in  land 
surveying  are  the  Chain  and  Tape  for  measuring  dis- 
tances, and  the  Picket,  Compass,  Solar  Compass  and 
Transit  for  running  lines. 

II.  INSTRUMENTS  FOR  MEASURING  DISTANCES  AND 
THEIR  USE. 

1.  The  Chain.    The  word  chain  is  used  to  represent 
a  distance  of  66  feet  and  also  an  instrument  used  for 
measuring  distances.    The   chain    in  most  general  use 
for   land  surveying   is    that   invented   by  Gunter,  and 
known  as  the  Gunter  chain.      It  is  66  feet  long    and 
divided  into  100  equal   parts,  called   links.    The  chain 
is  made  of  wire,  in  links  somewhat  less  than  eight  inches 
long.    These  are  joined  by    two    small,  round  or  oval 
rings  at  each  joint.    The  length  of  one  of  these  longer 
links,  with  the  two  rings  or  short  links  taken  together, 
make  the  distance  known  as  a  link. 

The  best  surveyor's  chains  are  made  of  steel 
wire,  having  the  links  brazed  to  prevent  stretch- 
ing by  opening  of  the  joints.  Chains  have  every  tenth 
link  marked  with  a  brass  tag.  The  tags  at  the 
end  of  the  tenth  link  from  each  end  have  one  point; 
those  at  the  twentieth  links  have  two  points;  those  at 
the  thirtieth  links  have  three  points;  those  at  the 
fortieth  links  have  four  points;  while  that  in  the  centre 
or  fiftieth  link  is  rounded  and  has  no  point.  Heavy 
chains  of  iron  wire,  with  open  joints,  are  of  little  value. 
It  is  very  difficult  to  measure  correctly  with  them,  over 
rough  ground,  owing  to  their  weight.  They  stretch 
rapidly  by  wear  and  by  the  opening  of  the  joints. 
Chains  fifty  links  long  are  used  to  measure  over  rough 
ground. 

2.  Chains  Stretch  by  use,  chiefly  from  wear  in  the 
joints.    The  best  steel  brazed  chains,  when  in  constant 
use  on  gritty  ground,  will  stretch  six  inches  or  more 
in  a  year  from  this  cause  alone.    They  may  be  corrected 
in  several  ways.    They  may  be  shortened  a  limited  amount 


INSTRUMENTS   FOR  MEASURING   DISTANCES.  O 

by  turning  up  the  nuts  or  burrs  which  hold  the  handles 
in  place.  They  may  be  shortened  by  taking  out  short 
links  or  rings.  The  better  way  is  to  distribute  the 
correction  evenly  throughout  the  chain,  by  putting 
each  link  in  a  vise  and  striking  lightly  on  the  end  with 
a  hammer,  shortening  it  in  that  way. 

The  links  in  the  chain  get  bent  by  use.  When  many 
of  them  are  bent,  the  chain  becomes  elastic  and  will 
elongate  from  one  to  two  inches  when  pulled.  Chains 
should  be  examined  before  using  and  the  links  straight- 
ened. They  should  be  frequently  compared  with  a  stand- 
ard, that  their  length  may  be  known,  and  they  should 
be  kept  near  the  true  length. 

3,  Steel  Tapes  are  made  for  the  use  of  land  sur- 
veyors. They  are  light,  so  that  they  may  be  readily  lev- 
eled up  in  measuring  over  rough  ground  or  on  a  slope. 
They  do  not  stre'tch.  There  are -no  links  to  get  kinked 
and  thus  cause  a  false  measure.  They  are  in  every  way 
more  accurate  and  convenient  than  the  chain.  The  best 
tapes  for  general  use  are  made  of  the  best  quality  of 
steel  ribbon,  polished  and  blued,  from  ^  to  %  of  an  inch 
wide,  and  No.  30  to  32  thick.  The  wider  thinner  tapes 
are  nearly  useless  for  field  work. 

Tapes  are  made  of  any  length  and  graduated  to  suit 
the  work  for  which  they  are  designed.  A  tape  66  feet 
long,  graduated  to  links,  is  best  adapted  to  country 
use.  Tapes  50  or  100  feet  long,  graduated  to  feet 
and  hundredths,  are  better  adapted  for  use  in  many  cities. 
Tapes  from  200  to  400  feet  long  or  even  longer  are  made 
for  special  uses.  With  them  long  lines  may  be  rapidly 
measured  with  an  accuracy  fairly  comparable  with  the 
best  work  of  the  coast  survey. 

Two  precautions  need  to  be  observed  with  steel  tapes. 
When  in  use  they  should  be  kept  out  at  full  length  and 
never  be  doubled  on  themselves.  If  doubled  they  are 
easily  kinked  and  broken.  When  done  up,  they  should  be 
wiped  clean  and  wound  on  open  reels  to  prevent  rusting. 


4  A  MANUAL   OF  LAND   SURVEYING. 

4.  A  light  wire  is  a  cheap  and  handy  substitute  for 
the  chain  or  tape.    It  is  necessary  to  find  its  length  in 
some  way  and  then  for  even  lengths  of  the  wire  it  is 
capable  of  as  accurate  work  as  the  best  tape. 

5.  Marking  Pins  are  used  with  the  chain  and  tape 
in  measuring.    They  are  usually  made  of  heavy  wire 
about  14  inches  in  length,  with  one  end  sharpened  to 
stick  in  the  ground  and  a  ring  turned  on  the  other  end 
for  convenience  in  handling.     Strips  of  cloth  are  tied 
in  the  rings  so  that   they   can   be  seen  more  readily. 
The  marking  pins  used  in  the  United  States  surveys 
have  heavy  points,  for  dropping  plumb  when  chaining 
on  slopes.    It  is  convenient  to  use  eleven  pins  in  chain- 
ing.    One  of  them  is  stuck   at   the  starting  point,  the 
leader  takes  ten,  and  then  there  is  always  one  to  start 
from,  when  the  tallies  are  kept  in  even  tens. 

6.  Measuring  or  chaining.    Two  men  are  required 
for  this,  and  a  third  man  can  be  of  great  assistance  when 
chaining  on  slopes  and  accurate  work  is  to  be  done. 
The  care   and   accuracy   required   will   depend  on  the 
interests  at   stake.    The   surveyor   would   mistake  his 
calling  who  should  attempt  to  measure  land  worth  fifty 
cents  an  acre  with  the  same  care  he  would  use  in  meas- 
uring land  worth  fifty    dollars   or  more  per  inch.    In 
making  measurements  the  following  things  are  to  be 
observed,  with    greater   or   less   care   and  accuracy  of 
detail,  according  to  the  importance  of  the  work  in  hand. 

1st.  Chains  are  not  adapted  to  great  accuracy  in  meas- 
urements. For  the  best  work  use  a  steel  tape,  of  which  the 
exact  length  at  a  given  temperature,  and  the  rate  of 
expansion  are  known.  Tapes  are  usually  made  to  be 
of  standard  length  at  a  temperature  of  about  60°,  F. 
The  rate  of  expansion  by  heat  varies  with  the  kind  and 
quality  of  steel  in  the  tape.  It  approximates  closely 
to  .000007  for  each  change  of  a  degree  in  temperature. 
Thus  a  tape  which  is  100  feet  long  at  60°  F.  will  be 
100.014  feet  long  at  80°  F.  For  very  exact  measurements 


INSTRUMENTS   FOR  MEASURING  DISTANCES.  5 

take  note  of  the  changes  in  temperature  and  correct 
for  expansion  and  contraction.  A  thermometer  is 
needed  for  this. 

2d.  Measure  in  straight  lines.  In  ordinary  work,  pick- 
ets or  rods  set  up  along  the  line,  in  sufficient  numbers 
for  the  chainmen  to  range  by,  will  enable  them  to 
secure  as  great  a  degree  of  accuracy  as  is  required  in 
this  respect. 

3d.  Measure  on  level  lines.  To  do  this  the  tape  may 
be  brought  to  a  level  line  and  the  successive  measures 
transferred  to  and  from  the  ground  by  plumb  lines. 
Use  a  plumb  having  a  fine,  strong  line  and  a  long,  well 
balanced,  sharp  pointed  bob.  Measure  down  the  slope. 
The  rear  chainman  should  hold  the  tape  steadily  and 
firmly  at  the  mark,  bracing  his  hand  against  his  leg  near 
the  ground  for  a  support.  The  leader  brings  his  end  of 
the  tape  level  and  in  line.  If  necessary  the  follower 
directs  him  in  doing  this.  He  then  applies  the  line  to 
the  point  or  mark  on  the  tape,  with  the  plumb-bob  very 
nearly  touching  the  ground.  When  he  has  the  proper 
tension  on  the  tape,  and  the  plumb  hangs  perfectly  still 
and  true,  he  depresses  the  line  enough  to  make  a  slight 
mark  on  the  ground  with  the  point  of  the  bob,  and 
sticks  his  marking  pin  beside  it. 

Another  method  of  getting  the  measure  on  level 
lines  is  to  drive  short  stakes  or  hubs  along  the  line  at 
every  change  in  the  slope  of  the  surface.  Small  headed 
tacks  are  driven  in  the  tops  of  these  hubs.  The  distance 
between  the  tackheads  is  then  measured  along  the  sur- 
face and  each  measurement  recorded.  A  level  is  then 
taken  showing  the  difference  in  hights  of  these  points. 
The  length  of  the  level  line  is  found  by  calculation. 
Between  every  two  hubs  we  have  a  right  triangle  in 
which  we  have  the  hypothenuse  given  by  the  tape,  and 
the  altitude  given  by  the  level,  to  find  the  base.  By  this 
method  the  error  may  be  reduced  below  1  in  25,000. 


6  A  MANUAL   OF   LAND   SURVEYING. 

4th.  The  tape  must  be  drawn  to  the  proper  tension. 
Tapes  are  usually  tested  under  a  tension  of  ten  pounds 
when  supported  the  entire  length.  They  should  be 
further  tested  to  find  the  amount  of  additional  strain 
required  to  overcome  the  sag,  when  the  tape  is  not 
supported  between  the  ends.  This  varies,  in  different 
tapes,  from  6  to  12  pounds  for  a  100  foot  tape.  The  total 
strain  in  the  unsupported  tape  in  measuring  should  be 
from  16  to  22  pounds.  The  exact  amount  is  to  be  found 
for  each  tape  by  trial. 

7.  The  following  is  the  general  method  of  procedure 
in  chaining,  modified  as  the  circumstances  require.  We 
will  speak  of  the  chainmen  as  leader  and  follower. 
The  leader  takes  his  end  of  the  chain  or  tape  and  ten 
marking  pins,  and  steps  briskly  in  the  direction  of  the 
line  to  be  measured.  One  pin  is  stuck  at  the  starting 
point.  Just  before  the  leader  has  the  chain  drawn 
out  at  full  length,  the  follower  calls  "halt,"  and  places 
his  end  of  the  chain  in  the  proper  position  at  the  start 
ing  point.  The  leader  shakes  out  any  kinks  there  may 
be  in  the  chain,  straightens  and  levels  it  in  the  line 
brings  it  to  the  proper  tension  and  sticks  his  pin,  calling 
"stuck"  when  he  has  done  so.  When  the  follower  hears 
this  signal,  and  not  before,  he  pulls  the  marking  pin 
and  both  move  quickly  forward,  repeating  the  opera 
tion  until  the  leader  has  stuck  his  last  pin  or  has 
reached  the  end  of  the  line.  When  the  leader  has 
stuck  his  last  pin  he  calls  "tally."  The  follower 
drops  his  end  of  the  chain  and  brings  forward  the  ten 
pins  which  he  has,  and  gives  them  to  the  leader,  who 
counts  them  to  be  sure  none  have  been  lost 
and  then  proceeds  as  before.  The  follower  need  not 
return  for  his  end  of  the  chain.  The  leader  will  draw 
it  forward  to  him.  When  the  end  of  the  line  is  reached 
the  leader  holds  his  end  of  the  chain  at  that  point 
while  the  follower  drops  his  end  and  comes  forward 
and  ascertains  the  distance,  if  any,  between  the  last  pin 
that  was  set  and  the  end  of  the  line. 


INSTRUMENTS   FOR  MEASURING   DISTANCES.  7 

When  chaining  on  slopes  which  are  so  steep  that  the 
whole  length  of  the  chain  cannot  be  leveled  at  once,  the 
leader  first  draws  it  forward  the  whole  length  and  in  the 
line.  He  then  drops  the  chain  and  all  his  marking  pins 
and  returns  to  a  point  where  he  can  level  a  part  of  the 
chain  and  measures  the  distance,  sticking  one  of  the  fol- 
lower's marking  pins  to  mark  the  point,  the  follower  then 
drops  his  end  of  the  chain,  comes  forward  and  taking  the 
chain  at  the  same  point  holds  it  to  the  mark  while  the 
leader  measures  a  second  section,  and  so  on  in  succession 
till  the  end  of  the  chain  is  reached,  where  the  leader 
sticks  one  of  his  own  marking  pins.  It  will  not  often 
be  necessary  to  take  any  note  of  the  lengths  of  the  parts 
of  the  chain  measured.  Observe  only  to  measure  to 
and  from  the  same  points  in  the  chain,  and  take  care  that 
the  count  is  not  lost  by  getting  the  marking  pins  im- 
properly mixed  together. 

The  follower  should  see  that  his  end  of  the  chain  is 
correctly  and  firmly  held  in  its  position  when  measuring. 
He  should,  when  necessary,  direct  the  leader  in  keeping 
the  true  line.  The  leader  should  see  that  his  chain  is  drawn 
straight,  level,  in  line,  and  to  a  uniform  tension.  To  assist 
him  in  keeping  the  line  he  should  observe  objects  in  the 
range,  both  front  and  rear.  He  should  see  that  his 
marking  pins  are  set  at  the  exact  point.  They  should 
either  be  set  plumb  or  slanting  at  right  angles  with  the 
line,  so  that  the  measure  may  be  taken  from  the  point. 
When  a  plumb  line  is  used,  the  latter  is  the  better  way. 
Chain  men  should  step  quickly  between  points,  and  in 
chaining  keep  up  with  a  man  walking  at  an  ordinary 
gait  of  three  miles  an  hour.  The  follower  must  not 
stop  the  leader  by  a  jerk  on  the  chain.  The  leader  must 
pull  steadily  when  measuring.  No  jerking  on  the 
chain  should  be  permitted^ 

If  there  is  a  difference  in  the  chainmen  the  best  man 
should  take  the  lead.  The  chaining  should  always  be 
uniform.  In  jnany  surveys  uniformity  of  measure  is 
more  important  than  great  exactness. 


8  4   MANUAL  OF  LAND   SURVEYING. 

Tests  made  by  the  author  have  led  him  to  the  conclusion,  that,  in 
common  country  surveying  with  the  chain,  nothing  is  gained  by  level- 
ing the  chain  where  the  ground  slopes  less  than  five  in  a  hundred.  He 
finds  that  in  field  practice,  under  the  ordinary  conditions,  more  is  lost 
by  the  sag  of  the  chain  than  is  saved  by  leveling.  In  one  careful  field 
test,  six  links  was  lost  in  a  mile  by  leveling  the  chain,  that  being  the 
net  difference  in  favor  of  surface  measurements  for  that  distance. 

In  that  class  of  work,  measurements  made  along  the  surface  may  be 
corrected  on  the  ground,  as  follows: 

When  ground  slopes  4  in  100  add  .1  link  per  chain 


8.  The  student  should  practice  in  the  field  with  the 
chain  and  steel  tape  until  he  is  entirely  familiar  with 
their  use,  and  can  do  accurate  and  rapid  work.  He 
should  measure  between  fixed  points  over  sloping  or 
uneven  ground,  and  repeat  the  measures  until  he  can 
secure  uniform  results.  He  may  be  surprised  at  first 
•to  tind  that  he  does  not  measure  twice  alike.  It  is  well 
to  drive  a  small  wooden  stake  at  every  tally  or  tenth 
chain,  so  that  in  case  a  marking  pin  is  lost  it  will  not 
be  necessary  to  go  back  farther  than  to  the  first  stake 
to  remeasure.  Beware  of  errors  in  counting  the  links 
less  than  a  full  chain.  Count  from  the  right  end  of 
the  chain  or  tape.  When  the  chain  is  used  do  not  mis- 
take the  tag,  as  60  instead  of  40  or  vice  versa,  or  count  odd 
links  the  wrong  way  from  the  tag.  Beware  of  such  mis- 
takes as  64  instead  of  56,  or  48  instead  of  52.  The  tape  is 
generally  numbered  the  whole  length  from  0  to  100, 
Nearly  the  same  care  is  needed  to  avoid  mistakes  in  read- 
ing as  with  the  chain,  especially  to  read  the  distance  from 
the  right  end  of  the  tape.  Otherwise  such  mistakes 
as  giving  the  distance  56  instead  of  44  are  very  liable, 
to  occur. 

III.  MEASURES  OF  LENGTH  AND  AREA. 

1.  The  measures  in  most  general  use  among  surveyors 
are  based  on  the  Gunter  chain.  The  surveyor  is  how- 
ever frequently  required  to  express  his  measurements  in 
units  of  the  old  linear  and  square  measure. 


MEASURES   OF  LENGTH   AND  AREA. 


9 


Table  of  Chain  Measure. 

7.92  inches  or  .66  foot=l  link. 

66  feet— 100  links=l  chain. 

80  chains=l  mile. 

In  country  surveying  the  smaller  measures  are  taken 
in  links  and  parts  of  a  link  and  distances  less  than  a 
quarter  of  a  link  are  not  counted  In  the  more  exact 
work  in  cities,  the  foot  and  its  subdivisions  are  in  com- 
mon use,  and  on  account  of  the  greater  ease  in  making 
computations  upon  the  decimal  system,  the  plan  of 
subdividing  the  foot  decimally  is  adopted  by  many 
surveyors,  and  is  growing  in  favor 

2.    Old  Linear  Measure: 
12  inches  =  1  foot. 
3  feet  =  1  yard. 
16^  feet  =  1  rod. 
40  rods  =;=  1  rood  or  furlong. 
320  rods  =  1  mile. 

MEASURES  FOR  AREA 

3    Chain  Measure: 
100,000  square  links,  or  >  _  t  _.„ 
10  square  chains      $  - 

640  acres  =  1  sq  mile  or  section. 
36  sections  —  1  township. 

In  the  United 
States  land  sys- 
tem, the  square 
mile  is  known  as 
the  Section.  It  is 
subdivided  into  al- 
iquot parts,  which 
are  described  ac- 
cording to  their 
place  in  the  sec- 
tion. The  manner 
of  naming  these 
subdivisions  of  a 
section  is  indicat- 
ed in  Figure  1. 


W  E  5  T 


FIG.  1. 


10 


A  MANUAL  OF  LAND  SUKYEYING. 


When,  because  of  lakes,  rivers,  reservations,  adjacence 
to  township  boundaries,  or  other  causes,  any  of  the  parts 
of  a  section  are  increased  or  diminished  from  their 
normal  amount,  they  are  known  and  described  as  Frac- 
tional. That  word  is  used  to  indicate  that  the  tract  to 
which  it  is  applied  is  not  one  of  the  regular  subdivisions 
of  the  section.  When  a  fractional  lot  is  small  it  is  the 
custom  of  the  United  States  land  department  to  attach 
it  to,  and  sell  it  with,  an  adjacent  larger  tract  which  gives 
the  name  to  the  description  of  the  whole  tract.  The 
manner  of  describing  fractional  lots  is  indicated  in  Fig- 
ure 2.  It  is  also  a  custom  to  number  the  fractional  lots 

on  the  plats  and 
describe  them  by 
numbers,  as  for 
.example,  Lot  No. 
3  of  Section  18. 
The  latter  method 
requires  a  refer- 
ence to  the  plat  to 
know  the  location 
of  the  lot,  while 
the  former  method 
does  not. 


4.  Old  English  Land  Measure: 

144  square  inches  =  1  square  foot. 
27234  square  feet  =  1  square  rod. 
40  square  rods  =  1  rood. 
160  square  rods  =  1  acre. 

Square  rods  and  feet  are  still  in  common  use  as  sub- 
divisions of  the  acre.  The  rood  and  furlong  are  very 
nearly  if  not  quite  obsolete  in  the  United  States. 

5.  Spanish   Measures. —  In  Spanish  colonies  in 
America,  the  Spanish  system  of  land  measures  was  used 


MEASURES   OF  LENGTH   AND   AREA.  11 

in  describing  and  measuring  the  land  grants,  and  has 
continued  in  use  down  to  the  present  time  in  a  large 
extent  of  country.  The  principal  unit  of  measure  is  the 
'  vara,"  which  seems  to  be  a  somewhat  variable  one.  In 
a  report  of  the  14th  of  November,  1851,  from  the  surveyor- 
general  of  California,  it  is  stated  that  all  the  grants,  etc., 
of  lots  or  lands  in  California,  made  either  by  the  Spanish 
government  or  that  of  Mexico,  refer  to  the  "vara"  of 
Mexico  as  the  measure  of  length;  that  by  common  con- 
sent, in  California,  that  measure  is  considered  as  exactly 
equivalent  to  thirty-three  American  inches.  That  officer 
enclosed  a  copy  of  a  document  he  had  obtained  as  being 
an  extract  of  a  treaty  made  by  the  Mexican  government, 
from  which  it  would  seem  that  another  length  is  given  to 
the  "vara;"  and  by  J.  H.  Alexander's  (of  Baltimore)  Dic- 
tionary of  Weights  and  Measures,  the  Mexican  vara  is 
stated  to  be  equal  to  .92741  of  the  American  yard.  The 
general  land  office,  however,  has  sanctioned  the  recog- 
nition, in  California,  of  the  Mexican  vara  as  being 
equivalent  to  33  American  inches. 

Extract  of  a  treaty  made  with  the  Mexican  government,  which  accom- 
panied a  report  dated  November  14,  1851,  from  the  U.  S.  surveyor- 
general  of  California,  respecting  the  ratio  of  land  measures  heticeen 
those  employed  under  the  Mexican  government  and  those  in  use  in 
the  United  States 

[From  the  Mexican  ordinance  for  land  and  sea.] 

Article  20th  of  the  agreement  entered  into  between  the  minister  pleni- 
potentiary of  the  Mexican  government  and  her  agents  in  London, 
the  15th  of  September,  1837,  with  the  holders  of  Mexican  bonds. 
20th.     In  compliance  of  what  is  ordered  by  the  seventh  article  of 
the  preceding  law,  and  in  order  to  carry  into  effect  the  stipulation  in 
the  preceding  agreement  in  regard  to  the  holders  of  bonds  deferred, 
it  is  declared  that  the  act  of  which  mention  is  made  in  said  agreement 
answers  to  4840  English  yards  squared,  equivalent  to  5762.403  Mexican 
varas  square;  inasmuch  that  the  "sitio  de  ganado  moyer"  contains 
4338.464  acres,  the  Mexican  vara  having  been  found  by  exact  measures 
equal  to  837  French  millimetres. 

Reducing  the  ratio  of  4840  square  yards 

and  5762.403  square  varas,  the  vara 

will  be 32.99312  inches 

Reducing  the  4338.464  acres 32.99311 


12 


A  MANUAL   OF  LAND  SURVEYING. 


2=3 

<n 

£ 

. 

1 

1 

1 

1 

Names  of  the 

Figures   of 

S  « 

a 

VI 

Measures. 

Measures. 

°l«i 

"Z 

C 

5  p,=3 

3 

tn  s3 

t».-3 

|rf 

**  >• 

I9 

Sitio  de  ganado  moyer 

Square  

5,000 

5,000 

25,000,000 

41.023 

Criadero    de    ganado 

moyer  _    _ 

do. 

2,500 

2,500 

6,250,000 

10.255 

Sitio  de  ganado  menor 

do. 

3,333  X3 

3,333% 

11  111  111 

18.232 

Criadero    de    ganado 

menor  

do.     _ 

1  666  2 

1,666% 

2,777  777g 

4.558 

Caballeria  de  tierra— 

Right  angled 

Media  caballeria 

l>n  rail  'gram 
Square 

1,104 

552 

552 
552 

609,408 
304,704 

1 

Cuarto     caballeria    o 

Suerte  de  tierra  

Right  angled 

Fenega    de     sembra- 

paralPgram 

552 

276 

152,352 

^ 

duro  de  maiz 

do 

276 

184 

50,784 

1-12 

Sala  para  casa 

Square 

50 

50 

2  500 

0  004 

Fundo  legal  para  pue- 

blos _      _    

do.     

1,200 

1,200 

1,440,000 

3.362 

The  Mexican  vara  is  the  unit  of  all  the  measures  of 
length,  the  pattern  and  size  of  which  are  taken  from  the 
Castilian  vara  of  the  mark  of  Burgos,  and  is  the  legal 
vara  used  in  the  Mexican  republic.  Fifty  Mexican  varas 
make  a  measure  which  is  called  "  cordel,"  which  instru- 
ment is  used  in  measuring  lands. 

The  legal  league  contains  100  cordels,  or  5,000  varas, 
which  is  found  by  multiplying  by  100  the  50  varas  con- 
tained in  a  cordel.  The  league  is  divided  into  two  halves 
and  four  quarters,  this  being  the  only  division  made  of  it. 
Half  a  league  contains  2,500  varas,  and  a  quarter  of  a 
league  1,250  varas.  Ancientty,  the  Mexican  league  was 
divided  into  three  miles,  the  mile  into  a  thousand  paces 
of  Solomon,  and  one  of  these  paces  into  five-thirds  of  a 
Mexican  vara;  consequently,  the  league  had  3,000  paces  of 
Solomon.  This  division  is  recognized  in  legal  affairs-  but 
has  been  a  very  long  time  in  disuse— the  same  as  the  pace 
of  Solomon,  which  in  those  days  was  called  vara,  and  was 
used  for  measuring  lands.  The  "  mark  "  was  equivalent 
to  two  varas  and  seven-eighths— that  is,  eight  marks  con- 


MEASURES  OF  LENGTH  AND  AREA.  13 

taining  twenty-three  varas— and  was  used  for  measuring 
lands,  i 

In  Texas  the  surveys  are  made  on  the  vara  system.  A 
20- vara  chain  is  used,  the  area  calculated  in  varas,  and 
when  necessary  reduced  to  acres.  The  field  notes  contain 
no  system  of  measurement  except  varas.  Nearly  all  the 
old  leagues  were  laid  off  in  rectangular  form,  and  nearly 
all  the  subdivisions  since  have  been  by  lines  parallel  with 
the  original  league  lines. 

The  following  table  of  comparisons  gives  the  system  of 
land  measures  in  use  in  that  state: 

1  vara  =  33^  inches. 
1900.8  varas  =  1  mile. 

25,000,000  sq.  varas  =  1  league  =  4428.4  acres. 
1,000,000  "        "     =1  labor  =  177.136     " 
5645.376  "     =  1     " 

1   "        "     =  .000177     " 


6.  Old  French  Measures  were  used  in  laying  off 
land  in  the  French  colonies,  and  still  find  a  place  in  some 
parts  of  the  country.  The  unit  was  the  "arpent,"  of 
which  there  were  different  values,  varying  from  three- 
fourths  of  an  acre  to  an  acre  and  a  half.  The  "  arpent 
d'ordonnance"  or  legal  arpent  equalled  1.262  acres,  and 
contained  100  square  perches  of  22  "pieds  du  roi"  on  a 
side. 

The  old  French  linear  measures  were  the  old  Paris  foot 
called  "pied  du  roi"  and  its  sub-multiples— 

12  points  =  1  ligne. 

12  ligne  =  1  ponce. 

12  ponce  =  1  pied  du  roi  =  12.789  inches. 
6  pieds  du  roi  =  1  toise,— interesting  as  being  the  unit 
employed  in  the  survey  of  the  great  French  meridian  arc, 
on  which  the  metre  was  founded. 

Modern  French  measures  are  upon  the  Metric  System. 


14:  A  MANUAL  OF   LAND  SURVEYING. 

7t    Standard  Measures. 

i 

The  constitution  of  the  United  States  says  that  con- 
gress shall  have  power  to  establish  a  system  of  weights 
and  measures.  It  has,  however,  never  done  so.  In  1832 
the  secretary  of  the  treasury  assumed  the  authority  to 
adjust  and  regulate  the  weights  and  measures  in  use  in 
the  custom  houses,  and  delegated  the  construction  and 
adjustment  of  standards  to  Mr.  Hassler,  who  was -then 
superintendent  of  the  coast  survey. 

The  standard  of  length  adopted  was  a  yard,  as  meas- 
ured between  the  27th  and  63rd  inches  of  a  scale  made  in 
London,  by  Troughton,  and  brought  to  this  country  in 
1814.  This  scale  is  a  copy  of  the  old  British  Standard, 
known  as  the  Bird  Standard  of  1760. 

At  a  temperature  of  59.62°  F.  it  is  equal  in  length  to 
the  Imperial  Standard  at  62°  F.  Although  Congress 
never  adopted  that  yard  as  a  standard,  it  authorized  the 
transmission  of  copies  thereof  to  the  several  states.  In 
many  of  the  states  these  copies  have  been  legally  adopted 
as  the  standards.  Other  states  have  no  legal  standards. 
The  Michigan  standard  is  a  brass  yard,  of  exact  length 
at  a  temperature  of  58.40°  F.  It  is  both  a  line  and  an  end 
measure.  It  is  doubtful  if  these  standards  in  the  several 
states  are  kept  in  such  a  manner  as  to  be  reliable  for 
purposes  of  comparison  or  if  they  are  so  kept,  whether 
the  officers  in  charge  of  them  have  the  skill  and  the 
facilities  required  for  making  accurate  comparisons. 
Standard  rods  are  sold  by  dealers  but  they  are  more  or 
less  discrepant  in  length.  Surveyors  who  desire  to  know 
the  true  length  of  their  standard  measures  can  send  them 
to  the  Superintendent  of  the  Coast  and  Geodetic  Survey, 
at  Washington,  who  will  cause  them  to  be  compared  and 
the  government  stamp  placed  on  them,  giving  their 
exact  length.  The  examination  and  test,  for  which  a 
fee  of  fifty  cents  is  charged,  secures  a  sufficient  degree  of 
accuracy  for  ordinary  purposes  of  the  surveyor.  Where 
an  extra  degree  of  accuracy  is  called  for  a  higher  fee  is 
charged. 


MEASUBES  OF  LENGTH   AND   AREA.  15 

Although  Congress  has  not  adopted  a  general  standard 
of  measure,  it  has  adopted  a  standard  for  the  measure- 
ment of  the  public  lands,  which  so  far  as  the  resurvey  or 
subdivision  of  those  lands  is  concerned  is  final.  In  sec- 
tion 2395  of  the  revised  statutes  of  the  United  States,  it 
is  enacted  that  "all  lines  shall  be  measured  with  chains 
containing  two  perches  of  sixteen  and  one-half  feet,  each 
subdivided  into  twenty  five  equal  links.  In  section  2396 
it  is  enacted  that  "All  the  corners  marked  in  the  surveys 
returned  by  the  Surveyor  General  shall  be  established  as 
the  proper  corners"  &c.;  and  that  "the  boundary  lines 
actually  run  and  marked  in  the  surveys  returned  by  the 
Surveyor  General,  shall  be  established  as  the  proper 
boundary  lines  of  the  sections  and  subdivisions  for  which 
they  were  intended,  and  the  length  of  such  lines  as  re- 
turned shall  be  held  and  considered  as  the  true  length 
thereof.'" 

This  enactment  makes  an  actual  standard  of  measure 
between  every  two  adjacent  corners  of  the  government 
survey,  which  is  the  only  legal  standard  for  measures  of 
that  line.  The  surveyor,  in  resurveying  or  subdividing 
the  public  lands,  has  thus  a  standard  laid  down  for  him 
on  ever^  line  previously  run  by  the  government  deputy 
surveyor  and  has  only  to  adjust  his  chain  to  that  stand- 
ard. This  is  practically  done  on  the  ground  by  apportion- 
ing any  difference  between  the  surveyor's  measure  of  a 
given  line  and  the  length  of  the  line  as  returned  in  the 
field  notes  pro  rata  between  its  different  parts. 

Example.— It  is  required  to  locate  the  half-quarter  cor- 
ner on  the  line  described  in  the  field  notes  as  running, 
"West  on  corrected  line  between  Sections  11  and  14 
39.72,  set  qr.  sec.  post,"  etc. 

Suppose  the  surveyor  on  measuring  this  line  finds  the 
distance  between  the  two  corners,  as  actually  marked  on 
the  ground,  to  be  by  his  chain  39.84  chains.  Then  his 
chain  is  too  short  and  its  legal  length  for  that  line  is  to 
its  nominal  length  as  39.72  is  to  39.84  and  the  distance  to 
the  half -quarter  corner  is  by  the  new  measure  19.92  chains. 


16  A  MANUAL  OF  LAND  SURVEYING. 

IV.    INSTRUMENTS  FOR  RUNNING  LINES  AND 
THEIR  USE. 

1.  The  instruments  most  commonly  used  in  running 
lines  are  the  picket,  the  compass  and  the  transit.    There 
are  various  modifications  of  the  compass  and  transit. 
The  methods  of  running  lines  with  these  instruments 
will  be  treated  of  in  connection  with  the  description 
of  them. 

2.  The  Picket  or  Rod  is  the  simplest  device  for 
ranging  lines.    It   is   simply   a   straight   rod   an   inch 
or  two  in  diameter  and  having  a  sharp  point  to  stick  in  the 
ground.    The  author  prefers  to  have  them  sharpened  to 
a  long  slim  point  at  the  top  also,  and  that  the  pickets  shall 
be  of  such  a  length  as  to  be  the  height  of  the  eye  when 
firmly  planted  in  the  ground.    Where  timber  is  plenty 
they  may  be  cut  from  small  straight  saplings,  or  split 
from  body  wood  as  they  are  wanted,  and  left  standing 
where  they  are  used,  as  a  guide  to  the  chainmen. 

3.  To  range  a  line  with  pickets.    Set  the  first  picket 
at  the  starting  point   and   a   second  a  short  distance 
away    in  the   direction  in  which   the    line   is   to   run. 
Then  go  ahead   and   set  picket   after   picket   at  such 
distances  apart   that   at   least   three   of  them  can  be 
distinctly  seen  at  the  same  time.    Set  the  pickets  plumb 
and  align  them  by  sighting  over  the  sharpened  points 
at  the  top.    A  plumb  line  will  be  of  assistance  in  rang- 
ing lines  over  uneven  ground.    Set  short  stakes  in  the, 
line  at  uniform  distances  apart.    Then  if  the  line  was 
intended  to  strike   a   particular   point   and  missed,  it 
may  be  corrected  by  measuring  the  perpendicular  dis- 
tance from  the  line  to  the  point,  and  then  moving  each 
intermediate  stake  its    proportional    part    of  that  dis- 
tance according  to  the  distance  it  is  from  the  starting 
point. 

Example  1.— Commencing  at  the  southwest  corner  of 
Mr.  B.'s  farm,  I  ran  north,  setting  stakes  on  the  trial 
line  every  ten  chains.  At  40.00  chains,  my  line  inter- 


INSTRUMENTS  AND   THEIR  USE.  17 

sected  the  north  line  of  his  farm  32  links  east  of  his 
northwest  corner.  What  correction  must  be  made  for 
each  stake  ? 

Solution. — The  first  stake  being  set  at  ^  the  distance 
between  points  must  be  corrected  %  of  32  =  8  links,  and 
as  the  trial  line  came  out  to  the  east  of  the  corner,  the 
stakes  on  that  line  must  be  moved  to  the  west.  The  2d 
stake  being  at  %  the  distance  between  points  must  be 
moved  west  ^  of  32  —  16  links.  Similarly  the  3d  stake 
must  be  moved  west  24  links. 

NOTE.— Sections  of  the  United  States  survey  are  tracts  of  one  mile 
square.  Monuments  are  set  at  each  corner  called  Section  Corners. 
Others  are  placed  midway  between  them  on  the  section  lines  called 
quarter  posts  or  quarter  section  corners.  Some  sections  greater  or 
less  than  these  are  called  Fractional  Sections. 

Example  2 — Commencing  at  a  point  12  links  west 
of  the  quarter  post  in  the  south  side  of  Section  20,  I 
ran  north,  setting  stakes  on  the  trial  line  every  ten 
chains.  At  80  chains  my  line  intersected  the  north  line  of 
the  section,  36  links  west  of  the  quarter  post.  What  cor- 
rection must  be  made  to  place  the  intermediate  stakes 
in  the  true  line  between  the  quarter  posts,  known  as 
the  quarter  line  ? 

Answer—  Commencing  with  the  first  ten  chain  stake 
tney  must  be  set  east,  15,  18,  21,  24,  27,  30,  and  33  links 
respectively. 

Example  3.— Commencing  at  a  point  24  links  west 
of  the  southwest  corner  of  section  16,  I  ran  a  trial 
line  north,  setting  stakes  every  ten  chains.  At  80.36 
chains,  the  line  intersected  the  north  line  of  the  section,  32 
links  east  of  the  section  corner.  What  is  the  correction 
to  be  made  at  each  stake  to  place  it  in  the  true  section 
line  and  at  the  equidistant  points  y  Answer  to  be  found 
by  the  student. 

NOTE.— This  solution  requires  corrections  both  for  line  and  meas- 
ure. It  is  a  cardinal  principle  of  land  law  that  the  original  measure- 
ments and  monuments  which  were  made  in  the  survey  in  accordance 
with  which  the  land  was  sold  are  in  law  the  true  measures  and  monu- 
ments. All  subsequent  measures  for  the  purpose  of  locating  bound- 
aries must  be  made  to  conform  with  the  original  measures. 
3 


18  A  MANUAL   OF   LAND   SURVEYING. 

Trial  or  random  lines,  as  they  -are  usually  called,  are 
often  run  one  side  of  the  true  line,  purposely  to  avoid 
obstacles,  like  fences  and  hedge  rows.  The  surveyor,  by 
a  judicious  selection  of  ground  for  the  random  line  can 
often  save  a  great  deal  of  labor  and  time  of  the  party, 
by  avoiding  obstacles  which  would  otherwise  have  to  be 
removed  or  offset  around.  Eandoms  from  which  the 
true  line  is  to  be  found  should  be  run  with  as  great  care 
as  any  line. 

The  student  should  practice  running  and  measuring 
trial  lines  between  points  until  familiar  with  the  pro- 
cesses. He  should  run  various  randoms  to  find  the  line 
between  the  same  points  and  see  how  they  agree  when 
corrected  for  true  line. 

4.  To  range  a  true  line  between  points  that  can  not  be 
seen  from  each  other  but  can  both  be  seen  from  some  inter- 
mediate point,  as  a  hill. 

Set  up  flags  at  the  two  points.  Two  persons  then 
take  pickets  and  station  themselves,  a  short  distance 
apart,  at  the  intermediate  position  from  which  the  flags 
can  be  seen.  They  face  each  other  and  each  in  turn 
aligns  the  other  between  himself  and  the  flag  toward 
which  he  faces,  until  the  true  line  is  reached,  when  the 
pickets  are  set  in  the  line. 

5.  To  pass  obstacles  in  the  line. 

From  the  last  two  pickets  preceding  the  obstacles,  set 
two  other  pickets  on  a  line  parallel  with  (the  true  line 
and  at  a  sufficient  distance  to  pass  the  obstacle.  Prolong 
the  parallel  line  far  enough  to  set  two  pickets  beyond 
the  obstacle  and  then  regain  the  original  line  by  meas- 
uring back  from  these  two  pickets. 

6.  The  methods  of  running  lines  with  the  compass 
and  transit  will  be  given  in  connection  with  the  descrip- 
tions of  these  instruments. 


DESCRIPTION  OF  INSTRUMENTS.  19 


.      CHAPTER  II. 

DESCRIPTION  OF  INSTRUMENTS. 

1.  The  Surveyor's  Compass.  The  essential  fea- 
tures of  the  surveyor's  compass  are  a  magnetic  needle  for 
finding  a  meridian  line,  a  circle  graduated  to  half  degrees 
.known  as  the  limb,  for  laying  off  angles  from  the 
meridian,  and  sights  attached  for  use  in  prolonging  lines 
on  the  ground. 

When  the  limb  and  sights  are  on  separate  plates  move- 
able  upon  each  other  around  a  common  center  through 
an  arc  of  15°  or  20°,  and  a  vernier  is  attached,  the  instru- 
ment is  known  as  the  Vernier  Compass. 

The  use  of  the  vernier  is  chiefly  for  setting  the  sights 
of  the  instrument  so  that  they  will  be  in  the  true  north 
and  south  line  when  the  magnetic  needle  points  to  zero 
on  the  limb.  There  is  only  a  small  portion  of  the  earth's 
surface  in  which  the  needle  points  to  the  true  north. 
A  lino  passing  through  those  places  where  the  needle 
points  truly  north  is  called  the  agonic  line  or  line  of  no 
variation.  This  line  runs  in  a  northerly  course  and  is 
constantly  changing  its  position.  At  all  places  outside 
the  line  of  no  variation,  the  needle  points  to  the  east 
or  west  of  true  north.  This  difference  between  the 
direction  of  the  needle  and  the  true  meridian  is  spoken 
of  as  the  variation,  or,  more  correctly,  the  declination 
of  the  needle.  The  vernier  is  used  to  measure  the  angle 
between  these  two  lines. 


A  MANUAL   OF  LAND   SURVEYING. 


FIG.  3.— VERNIER  COMPASS-6-INCH  NEEDLE. 

Sometimes  there  is  added  a  divided  circle  or  limb  with 
^erniers  by  which  angles  can  be  taken  throughout  the 
entire  circle  independently  of  the  needle.  The  instrument 
in  this  form  is  called  the  railroad  compass.  The  addition 
of  leveling  screws  and  a  revolving  telescope  in  place  of 
the  plain  sights  makes  a  surveyor's  transit  of  it. 


ADJUSTMENTS   OF   THE   COMPASS.  21 

The  Plain  Compass  consists  of  a  circular  box  of 
brass,  usually  about  six  inches  in  diameter,  resting  upon 
an  arm  of  the  same  metal  about  fourteen  inches  in  length- 
At  the  extremities  of  the  arm  are  vertical  attachments 
through  which  are  fine  slits,  terminated  at  intervals  by 
circular  apertures,  which  serve  as  sights  in  directing  the 
instrument  upon  any  point.  At  the  centre  of  the  box  is 
a  small  vertical  pin  upon  which  is  balanced  a  slender 
magnetized  bar  of  steel,  called  the  Needle. 

Turning  with  a  free  horizontal  motion,  the  pointed 
ends  of  the  needle  traverse  the  graduated  circumference 
of  the  circle.  The  plane  of  the  sights  passes  through  the 
center  of  the  circle  and  cuts  the  circumference  in  two 
points  marked  N  •  and  S,  otherwise  distinguished  as  the 
north  and  the  south  points  of  the  instrument.  From 
'  these  points  the  graduation  of  the  circle  runs  90°  in  each 
direction  to  the  points  marked  E  and  W. 

A  circle  of  plate-glass  forms  the  cover  of  the  box. 
Two  small  spirit  levels  are  placed  at  right  angles  to  each 
other  upon  the  arm,  to  aid  in  rendering  the  plane  of  the 
instrument  horizontal. 

The  compass  is  mounted  upon  a  three-legged  support 
called  a  Tripod,  or  upon  a  single  staff  called  a  Jacob 
Staff,  with  which  it  is  so  connected  as  to  admit  of  being 
turned  in  any  desired  direction.  In  using  the  compass, 
the*  surveyor  should  keep  the  south  end  toward  his  per- 
son, and  read  the  bearings  from  the  north  end  of  the 
needle.  He  will  observe  that  the  letters  E  and  W  on  the 
face  of  the  compass  are  reversed  from  their  natural 
position,  to  correspond  with  the  line  of  the  sights,  in 
order  that  the  direction  may  be  correctly  read. 

II.    ADJUSTMENTS  OF  THE  COMPASS. 

The  Sights  of  the  compass  should  be  truly  at  right 
angles  with  the  plate,  so  that  when  set  up  and  leveled 
ready  for  use  the  line  of  sight  will  be  in  a  vertical 
plane. 


22  A   MANUAL   OF   LAND   SURVEYING. 

The  needle  should  cut  opposite  degrees  in  any  part  of 
the  circle,  and  should  have  its  ends  in  line  with  the 
centre. 

The  levels  should  be  parallel  to  the  plane  of  the  plate. 
To  adjust  the  compass  to  these  conditions  begin  with 

The  Levels. — First  bring  the  bubbles  into  the  centre, 
by  the  pressure  of  the  hand  on  different  parts  of  the  plate, 
and  then  turn  the  compass  half-way  around;  should  the 
bubbles  run  to  the  edge  of  the  tubes,  it  would  indicate 
that  those  ends  were  the  highest;  lower  them  by  tight- 
ening the  screws  immediately  under,  and  loosening  those 
under  the  lowest  ends  until,  by  estimation,  the  error  is 
half  remo .  ed  ;  level  the  plate  again,  and  repeat  the  first 
operation  until  the  bubbles  will  remain  in  the  centre, 
during  an  entire  revolution  of  the  compass. 

The  Sights  may  next  be  tested  by  observing  through 
the  slits  a  fine  hair  or  thread,  made  exactly  vertical  by  a 
plumb.  Should  the  hair  appear  on  one  side  of  the  slit, 
the  sight  must  be  adjusted  by  filing  off  its  under  surface 
on  that  side  which  seems  the  highest. 

The  Needle  is  adjusted  in  the  following  manner: 
Having  the  eye  nearly  in  the  same  plane  with  the  grad- 
uated rim  of  the  compass-circle,  with  a  small  splinter  of 
wood  or  a  slender  iron  wire,  bring  one  end  of  the  needle 
in  line  with  any  prominent  division  of  the  circle,  as, the 
zero,  or  ninety  degree  mark,  and  notice  if  the  other  end 
corresponds  with  the  degree  on  .the  opposite  side  ;  if  it 
does,  the  needle  is  said  to  "cut"  opposite  degrees  ;  if  not, 
bend  the  centre-pin  by  applying  the  small  brass  wrench, 
furnished  with  the  compass,  about  one-eighth  of  an  inch 
below  the  point  of  the  pin,  until  the  ends  of  the  needle 
are  brought  into  line  with  the  opposite  degrees. 

Then,  holding  the  needle  in  the  same  position,  turn  the 
compass  half-way  around,  and  note  whether  it  now  cuts 
opposite  degrees ;  if  not,  correct  half  the  error  by  bend- 
ing the  needle,  and  the  remainder  by  bending  the  centre 
pin. 


ADJUSTMENTS  OF  THE  COMPASS.  23 

The  operation  should  be  repeated  until  perfect  rever- 
sion is  secured  in  the  first  position. 

This  being  obtained,  it  may  be  tried  on  another  quarter 
of  the  circle  ;  if  any  error  is  there  manifested,  the  correc- 
tion must  be  made  in  the  centre-pin  only,  the  needle 
being  already  straightened  by  the  previous  operation. 

When  again  made  to  cut^it  should  be  tried  on  the  other 
quarters  of  the  circle,  and  corrections  made  in  the  same 
manner  until  the  error  is  entirely  removed,  and  the  needle 
will  reverse  in  every  point  of  the  divided  surface.  If  the 
needle  has  lost  its  polarity,  and  needs  to  be  remagnetized, 
this  is  effected  in  the  following  manner  : 

The  operator  being  provided  with  an  ordinary  perma- 
nent magnet,  and  holding  it  before  him,  should  pass  with 
a  gentle  pressure  each  end  of  the  needle  from  centre  to 
extremity  over  the  magnetic  pole,  describing  before  each 
pass  a  circle  of  about  six  inches  radius,  to  which  the 
suriace  of  the  pole  is  tangent,  drawing  the  needle  towards 
him  and  taking  care  that  the  north  and  south  ends  are 
applied  to  the  opposite  poles  of  the  magnet. 

Should  the  needle  be  returned  in  a  path  near  the  mag- 
netic pole,  the  current  induced  by  the  contact  of  the 
needle  and  magnet,  in  the  pass  just  described,  would  be 
reversed,  and  thus  the  magnetic  virtue  almost  entirely 
neutralized  at  each  operation. 

When  the  needle  has  been  passed  about  twenty-five 
times  in  succession,  in  the  manner  just  described,  it  may 
De  considered  as  fully  charged. 

A  fine  brass  wire  is  wound  in  two  or  three  coils  on  the 
south  end  of  the  needle,  and  may  be  moved  back  or 
forth  in  order  to  counterpoise  the  varying  weight  of  the 
north  end. 

The  Centre- Pin.  — This  should  occasionally  be  ex- 
amined, and  if  much  dulled,  taken  out  with  the  brass 
wrench,  already  spoken  of,  or  with  a  pair  of  pliers,  and 
sharpened  on  a  hard  oil-stone — the  operator  placing  it  in 
the  end  of  a  small  stem  of  wood,  or  a  pin-visef  and  deli 


24  A   MANUAL  OF  LAND   SURVEYING. 

cately  twirling  it  with  the-  fingers  as  he  moves  it  back 
and  forth  at  an  angle  of  about  30  degrees  to  the  surface 
of  the  stone. 

When  the  point  is  thus  made  so  fine  and  sharp  as  to  be 
invisible  to  the  eye,  it  should  be  smoothed  by  rubbing  it 
on  the  surface  of  a  soft  clean  piece  of  leather. 

Electricity.— A  little  caution  is  necessary  in  handling 
the  compass  that  the  glass  covering  be  not  excited  by  the 
friction  of  cloth,  silk,  or  the  hand,  so  as  to  attract  the 
needle  to  its  under  surface. 

When,  however,  the  glass  becomes  electric,  the  fluid 
may  be  removed  by  breathing  upon  it,  or  touching  differ- 
ent parts  of  its  surface  with  the  moistened  finger. 

III.    To  RUN  A  LINE  WITH  THE  COMPASS. 

Set  up  the  instrument  at  the  point  from  which  the  line 
Is  to  run  ;  level  the  plate ;  turn  the  sights  in  the  direction 
in  which  the  line  is  to  run,  which  may  be  ascertained  by 
the  needle  or  otherwise,  as  is  most  convenient.  An  assist- 
ant, known  as  the  rodman  or  flagman,  goes  ahead  with  a 
sharp  pointed  rod  or  flag  pole  to  such  a  distance  as  is 
convenient,  and,  guided  by  the  signals  of  the  compass- 
man,  sets  his  rod  in  line.  When  the  ground  is  uneven,  the 
rodman  should  select  his  point  at  the  summit  of  rising 
ground,  when  possible  to  do  so,  in  order  to  save  unneces- 
sary setting  of  the  compass.  He  should  always  select  the 
point  most  favorable  for  setting  up  the  instrument,  both 
to  get  a  clear  spot  for  the  instrument  and  to  get  the  best 
point  for  taking  the  next  sight. 

When  setting  his  rod  he  should  face  the  compass,  hold- 
ing the  rod  plumb  and  directly  in  front  of  him.  He 
should  move  steadily  in  the  direction  indicated  by  the 
signals  and  not  stick  the  rod  down  until  he  receives  the 
signal  to  do  so.  After  sticking  it  he  should  look  for 
further  signals,  lest  a  change  in  its  position  might  be 
required.  After  the  rod  is  set  the  com  passman  should 
examine  his  instrument  to  see  that  it  is  in  position,  cor- 


TO   PASS   OBSTACLES  IN   THE   LINE.  25 

recting  it  and  resetting  the  rod  when  necessary.  He  then 
sets  up  a  picket  in  line  near  his  instrument,  to  be  used  for 
a  back  sight,  and  moves  his  compass  forward  in  the  line 
to  the  point  marked  by  the  rodman,  sets  it  up  in  the  line, 
with  the  sights  ranging  back  to  the  backsight,  and  con- 
tinues the  line  as  far  as  desirable.  The  needle  may  or 
may  not  be  used,  according  to  circumstances.  At  the 
beginning  of  the  line  the  direction  will  usually  be  obtain- 
ed from  the  needle.  If  used  afterwards  on  the  same  line, 
care  should  be  taken  to  have  it  in  proper  condition  and 
working  freely.  When  being  carried  the  needle  should 
be  raised  off  the  pivot,  otherwise  the  point  of  the  pivot 
will  become  dulled  and  the  needle  will  not  traverse  freely. 

IV.    To  PASS  OBSTACLES  IN  THE  LINE. 

1.  When  the  obstacle  is  a  tree,  and  no  great  degree  of 
accuracy  is  required,  make  a  mark  on  the  tree  where  the 
line  strikes  it  and  set  the  compass  up  on  the  opposite 
side  of  the  tree,  putting  it  in  line  by  taking  a  backsight 
on  the  tree,  and  finding  the  direction  of  the  line  by  the 
needle. 

2.  Make  an  offset  far  enough  to  pass  the  obstacle 
on  a  parallel  line,  the  same  as  when  running  a  picket 
line.    When  it  is  found  that  the  line  strikes  a  tree  too 
large  to  be  removed,  set  the  rod  in  line  near  the  tree,  and 
then  before  moving  the  compass,  set  the  picket  for  back- 
sight at  one  side  of  it,  a  sufficient  distance  to"  pass  the 
tree.    Then  move  the  compass  ahead  and  set  it  up  the 
same  distance,  and  direction  from  the  rod  that  the  back- 
sight picket  was  set  from  the  compass.    Get  the  direction 
of  the  line  by  ranging  to  the  backsight.    Prolong  the 
parallel  line  beyond  the  obstacle  and  regain  the  true  line 
in  a  similar  manner.    Other  methods  of  passing  obstacles 
in  line  will  be  given  further  on. 

Y.    THE  MAGNETIC  NEEDLE. 

1.    The  compass,  because  of  its  being  so  convenient  for 
use  has  been  for  many  years  the  principal  instrument  used 


26  A  MANUAL   OF   LAND   SURVEYING. 

in  Land  Surveying.  It  is  now  very  generally  superseded 
by  other  instruments  in  surveys  where  accuracy  is  re- 
quired. So  far  as  the  direction  of  lines  is  concerned,  all 
compass  surveying  is  based  on  the  tendency  of  the 
magnetic  needle  to  adjust  itself  to  the  magnetic  meridian 
when  free  to  do  so,  in  other  words  to  point  north  and 
south.  It  is  however  constantly  changing  its  direction. 

2.  Secular  Change.    The  line  of  no  variation,  as  it  is 
commonly  called,  otherwise  known  as  the  agonic  line 
seems  to  have  a  periodical  motion,  back  and  forth,  to  the 
east  and  west,  like  the  swinging  of  the  pendulum.    The 
length  of  the  period  is  unknown  but  probably  covers  sev- 
eral centuries. 

In  the  United  States,  so  far  back  as  known,  its  motion 
was  to  the  eastward  until  the  beginning  of  the  present 
century,  since  which  time  it  has  been  moving  westward. 
In  Michigan  the  secular  change  has  been  between  3' 
and  4'  per  year  to  the  westward  for  the  past  sixty 
years.  The  agonic  line  was,  iii  1890,  in  the  vicinity  of 
Lansing. 

3.  Diurnal  Change.    The   needle   when   undisturbed 
and  free   to   move,   swings   back    and  forth  each  day 
through  an  arc  varying  from  5'  to  2(K  or  more  in  amount. 
In  the  northern  hemisphere  the  north  end  of  the  needle 
moves  westward  from  about  8  A.  M.  until  about  1 :30  p.  M., 
then  returning  and  reaching  its  former  position  at  about 
8  P.  M.    The  amount  of  this  motion  is  not  uniform  from 
day  to  day,  being  least  on  cloudy  days  ;  nor  from  month 
to  month,  being  least  in  winter.    Nor  is  it  the  same  in 
different  localities.    The  effect  of  the  diurnal  variation  is 
such  that  if  a  surveyor  were  to  start  a  line  in  the  morning 
and  continue  running  it  all  day  in  the  same  direction,  as 
shown  by  the  needle,  he  would  run  a  line  like  a  letter  S. 

4.  Irregular  Changes.    The  needle  is  subject  to  sudden 
and  violent  changes  in  its  direction,  sometimes  coinci- 
dent with  a  thunderstorm  or  an  Aurora  Borealis,— often 
without  any  apparent  cause.    The  writer  has  observed  a 


THE  MAGNETIC  NEEDLE.  27 

Change  of  half  a  degree  in  less  than  ten  seconds  of  time, 
for  which  there  was  no  apparent  or  discoverable  cause. 
It  was  supposed  to  have  been  occasioned  by  a  magnetic 
storm. 

5.  Local  Attraction.    Iron  ore  in  the  earth,  or  iron  or 
steel  in  the  vicinity  of  the  needle  will  deflect  it  from  its 
normal  direction.    High  mountains  or  running  streams 
are  also  said  to  deflect  the  needle  more  or  less.    Pocket 
knives  and  steel  watch  chains  are  prolific  sources  of  error 
as  well  as  chains  and  axes. 

6.  Difference  in  Instruments.    It  is  found  by  obser- 
vation that  different  instruments  do  not  indicate  the 
same  declination  of  the  needle  when  observed  at  the 
same  time  and  place.    A  difference  of  15'  is  not  uncom- 
mon.   Eight  needles  of  three  types  made  at  Gurley's 
from  the  same  sheet  of  steel  and  tested  by  an  expert 
for  a  month  on  the  same  center  pin,  differed  in  direc- 
tion, and  the  difference  varied  with  the  time  of  day. 

7. ,  Things  to  be  Observed  in  Running  Compass  Lines. 
For  these  reasons  it  is  practically  impossible  to  run  a  true 
line  and  repeat  it,  relying  on  the  needle  alone  for  direc- 
1  ion.  Hence  in  ail  original  surveys,  made  with  the  com- 
pass, tte  field  notes  of  the  survey  should  give  the  date,  and 
state  whether  the  directions  of  the  lines  are  given  accord- 
ing to  the  magnetic  meridian.  If  not,  state  what  the 
angle  is  between  the  magnetic  meridian  and  the  meridian 
adopted  for  the  survey,  or  in  other  words  state  the  decli- 
nation of  the  needle,  estimated  or  allowed  for  in  the  sur- 
vey. The  meridian  adopted  will  usually  be  as  nearly 
coincident  with  the  true  meridian  as  known.  Back- 
sights should  be  used  whenever  the  line  is  prolonged 
beyond  a  single  sight,  both  to  secure  accuracy  in  the  line, 
and  as  a  check  against  local  disturbances  of  the  needle. 
They  also  save  time,  as  a  compass  can  be  pointed  to  a 
backsight  in  much  less  time  than  it  takes  a  good  needle 
to  settle. 


28  A   MANUAL   OF  LAND   SURVEYING. 

8.  Marking  Lines.  It  is  a  cardinal  principle  of  com- 
mon law,  as  well  as  the  statute  law  of  the  United  States 
with  reference  to  the  public  lands,  that  the  original 
surveys  as  marked  on  the  ground,  in  accordance  with 
which  the  land  was  sold,  are  conclusive  as  to  the  corners 
and  boundary  lines.  When  the  land  is  once  sold,  no 
change  can  be  made  in  the  marked  boundaries  without 
disturbing  the  vested  rights  of  the  owners.  Resurveys  are 
made  to  find  the  location  on  the  ground  of  the  original 
survey.  The  compass  is  a  useful  assistant  in  pointing 
out  where  to  look  for  the  more  certain  evidences,  such  as 
marked  trees,  stakes  or  corner  stones,  and,  in  the  absence 
of  anything  better,  may  be  used  to  determine  the  location 
of  the  line.  A  marked  tree  of  the  original  survey  is, 
however,  better  evidence  of  the  location  of  the  line  than 
any  line  afterward  run  by  a  compass.  It  is  possible  that 
the  line  might  be  exactly  retraced  by  the  compass,  but 
it  could  not  be  known  to  be  so  without  the  aid  of  other 
evidence.  Hence  the  marks  on  the  ground  which  define 
boundary  lines  cannot  be  made  and  kept  too  plain  and 
permanent.  The  field  notes  and  records  which  describe 
these  marks  should  be  full,  clear  and  concise. 

VI.    TRUE  MERIDIANS  AND  HOW  TO  FIND  THEM  WITH 
THE  COMPASS. 

In  a  country  that  has  had  the  first  surveys  made 
and  boundary  lines  marked,  and  subsequent  surveys  are 
based  on  these  lines,  it  is  very  rarely  of  any  consequence 
to  the  surveyor  to  know  where  the  true  meridian  is.  The 
original  boundary  lines  are  unchangeable,  and  it  is  no 
help  to  the  surveyor  to  know  where  the  true  meridian  is 
unless  he  also  knows  that  the  original  surveys  were  in 
conformity  with  it,  and  that  the  causes  of  error  hereto- 
fore mentioned  can  be  eliminated.  That  is  very  rarely  the 
case.  His  main  concern  is  to  know  where  the  lines  were 
and  not  where  they  ought  to  have  been.  The  writer  in 
nearly  a  quarter  century  of  active  practice  as  a  surveyor 
has  never  had  occasion,  except  as  a  matter  of  curiosity,  to 
know  where  the  true  meridian  was.  In  making  the 
surveys  of  a  country  with  a  compass,  it  is  well  to 


TO  FIND  A  TRUE  MERIDIAN. 


29 


know  the  position  of  the  true  meridian,  in  order  that  the 
lines  may  be  run  as  nearly  in  conformity  with  it  as  the 
limitations  of  the  instrument  will  permit,  or  that  the 
divergence  may  be  known.  Subsequently,  a  knowledge 
of  the  changes  in  the  declination  of  the  needle  is  all  that 
serves  any  practical  purpose.  This  can  be  learned  by 
observations  on  any  line  between  two  permanent  points. 

To  find  a  true  north  and  south  line  by  means  of  the 
north  star. 

The  north  star  appears  to  describe  a  small  circle  about 
the  true  north  point  or  pole  as  a  center.  The  radius  of 
this  circle  is  called  the  Polar  Distance  of  the  star. 
This  polar  distance  is  not  a  constant  quantity,  but  be- 
comes about  ^  of  a  minute  of  arc  less  every  year.  On 
the  first  of  January,  1890  it  was  about  1°  16'  41". 

When  in  its  revolution,  the  star  is  farthest  from  the 
meridian,  it  is  said  to  be  at  its  greatest  eastern  or 
western  elongation. 

The  times  of  the  elongations  as  given  by  a  correct 
clock,  for  latitude  from  38°  N  to  60°  N  and  for  the  year 
1890,  are  approximately  as  shown  in  the  following  tables: 

EASTERN   ELONGATIONS. 


Day. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

1 

7 
13 
19 
25 

6  37  A.M. 
614    " 
550    " 
526    " 
503    " 

439A.M. 
416    " 
352    " 
328    " 
305    " 

237A.M. 
214    " 
150    " 
126    " 
103    " 

1239A.M. 
121fr   " 
li  52  P.M. 
1*29    " 
1105    " 

10  37  P.M. 
1014    " 
950    " 
927    " 
903    " 

8  36  P.M. 
812    " 
748     * 
725    " 
701    " 

WESTERN   ELONGATIONS. 


Day. 

Oct. 

Nov.     I     Dec. 

Jan. 

Feb. 

Mar. 

II.  M. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

1 

13 
19 
25 

€  -fi  A.M. 
6W    " 
640    " 
617    " 
453    " 

425A.M. 
402     " 
338    " 
315    " 
251    " 

228A.M. 
204    " 
140    " 
117    " 
1253    " 

12  26  A.M. 
1202     " 
11  39  P.M. 
11  15    " 
10  51     " 

10  24  P.M. 
1000    " 
936     " 
913    " 
849    " 

8  30  P.M. 
806    " 
743    " 
719    " 
655    " 

30  A  MANUAL   OF  LAND   SURVEYING. 

To  find  the  meridian  of  a  place  by  means  of  an  elonga- 
tion of  the  north  star  requires  the  arrangement  of  the 
following  preliminaries. 

Set  two  posts  firmly  in  the  ground  about  three  feet 
apart  east  and  west,  and  saw  them  off  to  a  level  about 
three  feet  from  the  ground. 

Lay  upon  the  posts  a  plank  3  or  4  feet  long  and  6  or  8 
inches  wide,  planed  smooth  on  the  upper  surface,  and 
nail  or  pin  it  securely  to  the  supports,  forming  a  sort  of 
table. 

To  the  north  of  the  table  at  a  distance  of  10  or  12  feet 
set  in  the  ground  a  stiff  pole  12  or  15  feet  high,  having  a 
cross  bar  nailed  to  its  top,  in  an  east  and  west  direction, 
from  which  to  suspend  a  plumb-line  nearly  reaching  the 
ground,  and  having  a  bob  weighing  1  or  2  pounds,  which 
may  be  caused  to  hang  in  a  pail  of  water,  to  insure  stead- 
iness. 

Provide  also  a  block  or  piece  of  plank  8  or  10  inches 
long,  and  smooth  on  the  under  side.  Let  one  of  the  com- 
pass sights  be  fastened  at  right  angles  with  the  upper 
surface  of  the  block  and  even  with  the  side  which  is  to  be 
toward  the  south. 

Everything  being  in  readiness,  the  observer,  a  few 
minutes  before  the  time  of  an  elongation  as  given  in 
the  above  Table,  should  be  at  his  post  and  begin  moving 
the  block,  even  with  the  south  edge  of  the  table,  keeping 
the  plumb-line  and  star,  as  seen  through  the  vertical  slit, 
constantly  in  range  with  each  other.  A  light  will 
generally  be  needed  near  the  plumb-line,  to  render  it 
visible.  As  the  star  approaches  its  elongation,  it  will 
appear  to  move  nearly  vertical  for  several  minutes,  so  as 
to  be  seen  without  moving  the  sight.  When  it  is  certain 
that  the  star  has  reached  its  elongation,  confine  the  block 
carefully,  by  sticking  a  few  tacks  along  its  edges.  Pro- 
ject the  vertical  slit  to  the  ground  by  means  of  a  plumb- 
line  and  mark  the  point  by  setting  a  substantial  stake 
with  its  top  a  little  below  the  surface  of  the  ground. 


TO   FIND   A   TRUE   MERIDIAN.  31 

Being  still  careful  not  to  move  the  block,  let  an  assist- 
ant take  one  of  the  iron-pointed  rods,  or  a  stake,  with  a 
light,  and  go  a  hundred  feet  or  more  toward  the  star,  and 
having  found  the  point  as  directed  by  the  observer,  in 
range  with  the  plumb-line  as  seen  through  the  slit,  let 
him  mark  it  by  driving  a  stake. 

Having  now  two  stakes  in  range  of  the  elongation,  the 
remainder  of  the  operation  may  be  deferred  till  morning. 

To  find  the  angle  which  the  line  as  above  determined 
makes  with  the  meridian  of  the  point  of  observation, 
requires  a  trigonometrical  computation. 

Let  A  be  the  point  of  ob- 
servation, Z,  the  zenith  of 
that  point,  HO,  an  arc  of  the 
northern  horizon,  N,  the 
north  point  of  that  arc,  /S, 
the  north  star  at  its  eastern 
elongation,  PS,  the  polar 
distance  of  the  star,  A  N,  the 
meridian  of  the  point  of 
observation,  and  AE,  the  line  of  the  two  stakes. 

The  angle  sought  is  NAE  =  angle  PZS  =  arc  NE.  ' 

Now,  in  the  spherical  triangle  PZS,  PZ  is  the  co-latitude 
of  the  point  A,  which  must  be  known.  Solving  this  tri- 

sin  PS  sin  polar  dist. 

angle,  we  have  sin  Z  = ,  or  sin  Z  — 

sin  ZP  cos  lat. 

From  this,  the  angle  Z  becomes  known,  and,  accord- 
ingly, it  may  be  formed  on  the  west  side  of  the  line 
AE,  and  thus  the  direction  of  the  meridian  AN  deter- 
mined. 

Ou  AN,  thus  found,  let  a  substantial  stake  be  set  a  hun- 
dred yards  or  more  from  At  and  we  have  a  permanent 
meridian  with  which  we  irtay  compare  the  magnetic 
meridian  at  any  time,  and  thus  determine  the  declination 
of  the  needle. 


32 


A   MANUAL   OF   LAND   SURVEYING. 


The  declination  of  the  needle  is  the  angle  which  the 
magnetic  meridian  makes  with  the  astronomical  merid- 
ian. 

For  the  purpose,  simply,  of  finding  the  declination  of 
the  needle,  it  is  sufficient  to  lay  out  on  the  ground  the 
line  of  direction  of  the  star  at  one  of  its  elongations,  and 
then,  knowing  the  bearing  of  this  line  as  shown  by  the 
needle,  and  the  corresponding  azimuth  of  the  star,  the 
declination  of  the  needle  is  readily  computed. 

Thus,  let  ±  a  =  azimuth,  =b  6  =  bearing,  and  =fc  d 
=  declination,  accordingly  as  they  are  east  or  west. 

Then  ±  d  —  ±  a  —  (=b  6). 

RULE. — Subtract  the  bearing  from  the  azimuth. 

In  applying  the  Rule,  due  regard  is  to  be  had  to  the 
algebraic  signs. 

A  near  approximation  to  a  true  meridian  may  be  had 
by  observing  the  pole  star  while  it  is 
in  the  same  vertical  plane  with  the 
Bear.  star  Delta,  in  the  constellation  Cas- 
siopeia. When  both  are  behind  the 
plumb-line  together,  they  are  very 
nearly  in  the  true  meridian.  When 
Delta  Cassiopeia  passes  the  meridian 
above  the  pole,  it  is  too  high  in  the 
heavens  to  serve  this  purpose.  It 
passes  the  meridian  below  the  pole 
at  midnight  April  10th,  and  may  be 
used  for  two  months  before  and 
after  that  date.  Six  months  later 
the  star  Zeta,  the  last  but  one  in 
the  tail  of  the  Great  Bear,  takes  its 
*  *  *  place.  Fig.  5  shows  the  relative  po- 
a  *  sition  of  these  stars  and  the  pole, 

FIG,  5. 


Pal* 


VII.     OTHER    METHODS     FOB    FINDING    A    TRTTB 
MERIDIAN. 

There  are  various  other  methods  for  finding  a  true 
meridian,  several  of  which  are  here  given.  The 
Method  for  the  determination  of  the  azimuth  of 
Polaris  and  true  meridian  at  any  hour>  the  star 
being  visible,  and  the  correct  local  mean  time  known 
is  from  the  U.  S.  Surveying  Instructions. 

IN  this  article  it  is  proposed  to  present  a  method,with 
two  new  and  compact  tables  adapted  to  common  clock 
time,  with  such  plain  directions  for  use  that  any  person 
of  ordinary  intelligence  can  understand  and  apply  them. 

As  the  surveyor  should  have  a  perfectly  clear  idea  of 
what  is  meant  by  Astronomical  Time  (used  to  simplify 
computations),  and  the  Hour  Angle  of  .Polaris,  these 
terms  will  now  be  explained. 

The  Civil  Day,  according  to  the  customs  of  society, 
commences  at  midnight  and  comprises  twenty-four 
hours  from  one  midnight  to  the  next  following.  The 
hours  are  counted  from  12  to  12  from  midnight  to  noon, 
after  which  they  are  again  reckoned  from  12  to  12  from 
noon  to  midnight.  Thus  the  day  is  divided  into  two 
periods  of  12  hours  each;  the  first  of  which  is  marked 
a.  m.,  the  last  p.  m. 

The  Astronomical  Day  commences  at  noon  on  the  civil 
day  of  the  same  date.  It  also  comprises  twenty-four 
hours;  but  they  are  reckoned  from  0  to  24,  and  from  the 
noon  of  one  day  to  that  of  the  uext  following. 

The  civil  day  begins  twelve  hours  before  the  astro- 
nomical day;  therefore  the  first  period  of  the  civil  day 
answers  to  the  last  part  of  the  preceding  astronomical 
day,  and  the  last  part  of  the  civil  day  corresponds  to  the 
first  part  of  the  astronomical  day.  Thus,  January  9,  2 
o'clock  p.  m.,  civil  time,  is  also  January  9,  2h,  astro- 
nomical time;  and  January  9,  2  o'clock  a.  m.,  civil  time, 
is  January  8,  14h,  astronomical  time. 

Ttie  rule,  then,  for  the  transformation  of  civil  time 
Into  astronomical  time  is  this:  If  the  civil  time  is  marked 

107 


o4  A   MANUAL   OF   LAND   SURVEYING. 

p.  m.,  take  away  the  designation  p.  ra.,  and  the  astronomical 
time  is  haa  without  furtlwr  change ;  if  the  civil  time  is  marked 
a  m.,  take  one  from  the  day  and  add  twelve  to  the  hours,  re- 
move the  initials  a.  m.,  and  the  result  is  the  astronomical  time 
wanted. 

The  substance  of  the  above  rule  may  be  otherwise 
stated,  as  follows:  When  the  surveyor  takes  an  observa- 
tion during  p.  m.  hours,  civil  time,  he  can  say:  the  as- 
tronomical time  is  the  hours  and  minutes  passed  since 
the  noon  of  this  day,  and  when  observing  in  the  a.  m. 
hours,  he  can  say  the  astronomical  time  is  the  hours  and 
minutes  elapsed  since  the  noon  of  yesterday,  in  either 
<ase  omitting  the  designation  a.  m.  or  p.  m.,  and  writ- 
ing for  the  day  of  the  month,  that  civil  date  on  which 
the  noon  falls,  from  which  the  time  is  reckoned.  Fi- 
nally, the  astronomical  time  may  be  called  the  hours  and  min- 
utes elapsed  since  the  NOON  LAST  PASSED,  the  astronomical 
DATE  being  that  of  the  civil  day  to  which  the  noon  belongs- 
Thus,  April  23,  4:15  p.  m.,  civil  time,  is  April  23,  4h  15m, 
astronomical  time,  and  April  23,  4: 15  a.m.,  civil  time,  is 
April  22,  16h  15m,  astronomical  time. 

The  surveyor  should  thoroughly  master  this  trans- 
formation of  the  civil  time  into  astronomical  time,  as 
it  will  be  the  first  duty  he  will  have  to  perform  after 
observing  Polaris  out  of  the  meridian. 

The  change  can  always  be  made  mentally,  no  written  work  being  re- 
quired. Table  I  might  be  easily  altered  to  give  the  times  by  the  civil 
count  marked  a.  m.  and  p.  m.,  but  such  an  arrangement  would  greatly 
extend  and- complicate  the  following  -ule8  and  examples,  and  correspond- 
ingly increase  the  chances  for  making  mistakes. 

Hour  Angk  of  Polaris.—  In  Fig.  2,  Plate  I,  the  full  ver- 
tical line  represents  #  portion  of  the  meridian  passing 
through  the  zenith  Z  (the  point  directly  overhead),  and 
intersecting  the  northern  horizon  at  the  north  point  N, 
from  which,  for  surveying  purposes,  the  azimuths  of 
Polaris  are  reckoned  east  or  west.  The  meridian  is 
pointed  out  by  the  plumb  line  when  it  is  in  the  same 
plane  with  the  eye  of  the  observer  and  Polaris  on  the 
meridian,  and  a  visual  representation  is  also  seen  in  the 
vertical  wire  of  the  transit,  when  it  covers  the  star  on 
the  meridian,, 


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V 

A 


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VJ- 


h\  .u 

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& 

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r.  O  ^    >  ^O  Mfcff  UALt  #P  «iAJK  D  SURVEYING. 


it  is  said  to  cul- 
minate; above  the  pole  (at  S),  the  passage  is  called  the 
Upper  Culmination,  in  contradistinction  to  the  Lower 
Culmination  (at  S'). 

In  the  diagram,  —  which  the  surveyor  may  better  un- 
derstand by  holding  it  up  perpendicular  to  the  line  of 
sight  when  he  looks  toward  the  pole,—  Polaris  is  sup- 
posed to  be  on  the  meridian,  where  it  will  be  about  noon 
on  April  10th  of  each  year.  The  star  appears  to  revolve 
around  the  pole  in  the  direction  of  the  arrows,  once  in 
every  23h  56m  4s.  09  of  mean  solar  time  ;  it  consequently  comes 
to  and  crosses  the  meridian,  or  culminates,  nearly  four 
minutes  earlier  each  successive  day.  The  apparent  mo- 
tion of  the  star  being  uniform,  one  quarter  of  the  circle 
will  (omitting  fractions)  be  described  in  5h  59m,  one  half 
in  llh  58m,  and  three  quarters  in  17h  57m.  For  the  posi- 
tions S1}  s2,  s8,  etc.,  the  angles  SPsn  SPsa,  SPs3,  etc., 
are  called  Hour  Angles  of  Polaris  for  the  instant  the  star 
is  at  SD  S2,  or  s3,  etc.,  and  they  are  measured  by  the  arcs 
Ss15  Ss2.,  Ss3,  etc.,  expressed  (in  these  instructions)  in  mean 
solar  (common  clock)  time,  and  are  always  counted  from 
the  upper  meridian  (at  S),  to  the  west,  around  the  circle 
from  Oh  Om  to  23h  56m.l,  and  may  have  any  value  between 
the  limits  named.  The  hour  angles,  measured  by  the 
arcs  Sslf  Ss2,  Ss3,  Ss4,  Ss5,  and  Ss6,  are  approximately 
i*  8m,  5h  55m,  9h  4m,  14h  52%  18h  Olm,  and  22h  48m  respect- 
ively;  their  extent  is  also  indicated,  graphically,  by 
broken  fractional  circles  about  the  pole.  The  hour  an- 
gle, 5h  55m  and  18h  Olm  are  those  at  west  and  east  elonga- 
tion, respectively,  in  latitude  40°  N. 

Suppose  the  star  observed  (e.  g.)  at  the  point  S3;  the 
time  it  was  at  S  (the  time  of  upper  culmination),  taken 
from  the  whole  circle,  23h  56m.l,  will  leave  the  arc  Ssj, 
S2,  s3,  or  the  hour  angle  at  the  instant  of  observation; 
similar  relations  will  obtain  when  the  star  is  observed 
in  any  other  position;  therefore,  in  general:  — 

Subtract  the  time  of  Upper  Culmination  from  the 
correct  local  mean  time  of  observation;  the  remainder 
will  be  the  Hour  Angle  of  Polaris. 

The  observation  will  be  made  as  heretofore  directed, 


TO  FIND 

DEPARTMENT  OF  CIVIL  KMClMCCMINC 

modified  as  follows  :  TheT^Wff^fe#*oC^fctfift«W*%i( 
star  to  reach  elongation;  the  observation  may  be  made 
at  any  instant  when  Polaris  is  visible,  the  exact  time 
being  carefully  noted. 

TABLE  V. 

This  table  gives,  in  "  Part  I,"  the  local  mean  time  of 
the  upper  culmination  of  Polaris,  on  the  1st  and  loth  of 
each  month,  for  the  years  1901  to  1910,  inclusive.  The 
times  decrease,  in  each  year,  to  April  10,  when  they  be- 
come zero;  then,  commencing  at  23h  56m.l,  the  times 
again  decrease  until  the  following  April,  and  so  on,  con- 
tinuously. The  quantity  in  the  column  marked  "  Diff. 
for  1  day  "is  the  decrease  per  day  during  the  interval 
of  time  against  which  it  stands,  and  answers  for  att  the 
years  marked  in  the  table.  For  any  intermediate  date, 
the  "Diff.  for  1  day"  will  be  multiplied  by  the  days 
elapsed  since  the  preceding  tabular  date,  and  the  prod- 
uct subtracted  from  the  corresponding  time,  to  obtain 
the  required  time  of  upper  culmination  for  the  date 
under  consideration.  The  table  answers  directly  for 
108°  west  longitude.  The  results  of  using  it  for  other 
longitudes  will  contain  an  amount  of  error  hardly  ap- 
preciable, as  the  correction  for  longitude  cannot  exceed 
one-tenth  of  a  minute  of  time  for  each  9  degrees  of 
longitude.  A  few  examples  will  illustrate  the  use  of 
the  table. 

1.  Required  the  time  of  upper  culmination  of  Polaris  for  a  station  in 
longitude  116°  west,  for  March  3,  1904. 

h.  m. 

Astron.  time,  U.  C.  of  Polaris,  1904,  March  15 151.9 

Red.  for  12  days  is  3».94xi2=47-.3,  add. 47.3 

Local  mean  time,  U.  C.  of  Polaris,  1892,  March  3 2  39.2 

The  required  time  may  also  be  obtained  by  using  the 
table  in  the  opposite  direction;  by  taking  the  time  for 
March  15,  and  adding  the  reduction,  as  follows: — 

h.  m. 

Astron.  time,  U.  C.  of  Polaris,  1904,MaT"hl 2  47.0 

Red.  for  2  days  is  3m.94x2=7».9  (Part  H)  Snbtract 7.9 

Local  mean  time,  U.  C.  of  Polaris,  1904,  March  3 2  89.1 


QS  A   MANUAL  OF  LAND  SURVEYING. 

d5 

In'  this  case  the  two  results  are  identical.  If  the 
computation  is  made  both  ways,  the  results  will  check 
each  other. 

Part  II  has  been  inserted  to  save  the  surveyor  the  lit- 
tie  trouble  of  making  multiplications ;  thus,  for  the 
above  example,  look  in  Part  II,  under  the  proper  tabu- 
lar difference,  3m.94,  and  opposite  the  3d  or  17th  day 
of  the  month  in  left-hand  column  is  the  correction  7m.9. 

Computing  from  a  preceding  date,  for  days  between 
April  11  and  15  of  any  year,  the  reduction  in  Part  II 
will  be  greater  than  the  tabulated  time  of  culmination, 
in  which  case  23h  56m.l  will  be  added,  to  make  the  sub- 
traction possible. 


2.  Required,  for  a  station  in  long.  90°  west,  the  time  of  U.  C.  of  Polaris 
for  April  14,  1906— 

h.  m. 

Astrou.  time,  U.  C.  of  Polaris,  1906,  April  1.  (Part  I) 0  47.9 

Add 2356.1 

Sum 2444.0 

Reduction  to  April  14  (Part  II;,  subtract 51.1 

Local  mean  time,  U.  C.  of  Polaris,  April  14 23  52.9 


Working  from  the  following  date,  for  days  between  the 
9th  and  15th  of  April,  the  sum  will  exceed  23h  56m.l, 
and  when  this  occurs  subtract  23h  56™.  1  from  the  sum, 
and  the  remainder  will  be  the  required  time. 


3.  Required,  for  a  station  in  long.  90°  west,  the  time  of  U.  C.  of  Polaris 
for  April  10,  1903— 


h.  m. 

Astron.  time,  U.  C.  of  Polaris,  1903,  April  15  (Part  I) 23  48.5 

Reduction  for  5  days  (Part  H),  add 19.6 

Sum 2408.1 

Subtract ,28  56.1 

Local  mean  time,  U.  C.  of  Polaris,  1903,  April  10 0  12.0 


TO   KIND  A  TRUE   MERIDIAN. 


39 


V.— Local  mean  (astronomical)  time  of  the  upper  culmination  of  Poiarit, 
computed  for  longitude  108"  (7ft.  12m.)  west  of  Greenwich. 


Part  I. 

Dau. 

1801. 

1902. 

1903. 

1904. 

1906. 

1906. 

1907. 

1908. 

Diff.  for 
1  day. 

Jan.      1 

h.    m. 
6  39.5 

4.     m. 
6  41.0 

6  42.  4 

6  43.9 

641.4 

642.8 

644.3 

6  46.7 

3.96- 

16 

6  44.2 

6  45.7 

6  47.1 

6  48.6 

6  46.1 

647.6 

649.0 

6  50.4 

3.96 

Fab.       1 

4  37.1 

4  38.6 

4  40.0 

4  41.5 

4  39.0 

4  40.4 

4  4L9 

443.3 

3.96 

15 

3  41.3 

3  43.4 

3  44.8 

346.3 

3  43.8 

3  46.2 

3  46.7 

3  48.1 

3.96 

UK.     I 

246.6 

248.1 

2  49.5 

8  47.0 

8  48.5 

2  49.9 

2  51.4 

2  48.9 

3.H 

1  51.5 

1  63.0 

1  64.4 

1  61.9 

1  63.4 

1  64.8 

1  66.3 

1  53.8 

3.94 

Apr.      1 

0  44.6 

0  46.1 

0  47.6 

0  45.0 

046-.5 

047.9 

0  49.4 

0  46.8 

3.94 

21  45.6 

23  47.1 

33  48.6 

23  46.0 

23  47.5 

8348.9 

23  60.4 

2347.8 

3.93 

May     1 

22  42.8 

22  44.3 

22  46.7 

22  43.2 

22  44.7 

88  46.1 

22  47.6 

2846.1 

3.93 

r 

21  47.9 

21  48.3 

21  50.7 

81  48.2 

21  49.7 

81  61.  1 

21  62.6 

21  60.1 

3.92 

Jane     1 

2041.2 

2042.7 

1044.1 

8041.6 

2043.1 

2044.5 

tO  46.0 

20  43.6 

3.92 

15 

19  46.4 

19  47.9 

19  49.3 

1946.8 

19  48.3 

19  49.7 

19  61.2 

19  48.7 

3.91 

July      1 

13  43.8 

18  46.3 

18  46.7 

1844.2 

18  46.7 

18  47.  I 

18  48.6 

18  46.1 

3.91 

15 

17  49.0 

17  60.1 

17  51.9 

17  49.  4 

17  60.9 

1758.3 

17  53.8 

17  61.3 

3.92 

Aug.      1 

16  42.4 

16  43.9 

16  46.3 

16  42.  8 

16  44.3 

16  46.7 

16  47.  2 

16  44.7 

3  92 

15 

1647.6 

15  49.1 

1660.6 

1548.0 

16  49.5 

15  60.9 

15  52.4 

15  49.9 

3.92 

Sept.      1 

1441.0 

1442.5 

14  43.9 

14  41.4 

14  42.9 

14  44.3 

14  46.8 

14  43.3 

3.92 

15 

13  46.1 

13  47.  6 

13  49.0 

1346.6 

1348.0 

1349.4 

13  50.9 

13  48.4 

3.93 

Oct.       1 

12  43.3 

12  44.8 

12  46.2 

1843.7 

12  45.8 

1246.6 

12  48.1 

12  46.6 

3.93 

15 

11  48.3 

11  49.8 

11  61.2 

11  4-7 

11  60.2 

11  61.6 

H  53.1 

11  60.6 

3.93 

Nov.     1 

10  41.4 

10  42.9 

10  44.3 

10  41.8 

1043.3 

10  44.7 

1046.2 

1043.7 

3.«3 

16 

9  46.4 

947.9 

9  49.3 

9  46.8 

9  48.3 

9  49.7 

9  51.2 

9  4S.7 

3.94 

Dx.      1 

8  43.3 

8  44.8 

8  46.2 

8  43.7 

8  46.2 

8  46.6 

8  48.1 

8  45.6 

3.94 

15 

7  48.1 

7  49.6 

7  61.0 

748.6 

7*0.0 

7  51.4 

7  62.9 

7.60.4 

3.95 

Part  I—  Continued. 

Part  II. 

Date. 

1909. 

1310. 

1911. 

Difl. 
for 

Seduction  of  tabular  tine*  to  intermtdioU  dolt,. 

Iday. 

Subtract  the  reduction  when  competing  from  *\  preceding 

Jtn      1 

ft.     m. 
6  43  2 

h.     m. 
6  44  7 

*.    m. 
6  45  1 

3%5 

or  add  it  when  working  from  •.fotlowiny  date. 

15 

5  47.9 

5  43.'  4 

5  Mis 

a.  05 

Bednction.     Arg.  "Diff.  forl  day  " 

IVb.       1 

4  40.8 

4  42.3 

4  43.7 

3.03 

Day  of 

No.  of 

15 

3  46.6 

3  47.1 

348.6 

3.95 

the 

day* 

M..r.      1 

2  60.3 

2  C1.8 

2  63.2 

3.94 

month. 

m. 

M. 

m. 

w> 

j^ 

15 

1  65.2 

1  63.7 

1  58.1 

3.94 

3.91. 

3.92. 

3.93. 

3.94. 

3.96. 

eiapeeo. 

Ayr.     1 

048.3 

0  49.8 

0  51.2 

3.94 

15 
mr.  •       i 

23  49.3 
22  46  5 

2360.8 
22  48  0 

2352.2 
22  49  4 

8.93 
3  93 

7  j 

21  5L6 

21  63!  0 

21  54^4 

3.98 

or  16 

3.9 

3.9 

3.9 

3.9 

3.9 

Jon*     1 

20  44.9 

20  46.4 

20  47.8 

3.98 

or  17 

7.8 

7.8 

7.9 

7.9 

7.9 

16 

19  60.1 

19  61.6 

19  63.0 

3.91 

or  18 

1LY 

11.8 

11.8 

11.8 

n  • 

July     1 

18  47.6 

18  49.0 

18  60.4 

3.91 

or  19 

15.6 

16.7 

15.7 

15.8 

16.8 

16 

17  £2.7 

1764.2 

17  66.6 

a.  62 

or  20 

19.6 

19.6 

19:  6 

19.7 

19.7 

Aug.     1 

16  4S.1 

1647.6 

16  49.  0 

3.92 

or  21 

83.6 

23.5 

83.6 

83.6 

23.7 

15 

15  61.3 

15  62.  8 

1564.2 

S.»2 

or  22 

27.4 

27.4 

27.6 

27.6 

27.6 

Sept.     1 

14  44.7 

14.46.2 

1447.6 

3.92 

9  or  23 

31.3 

31.4 

31.  i 

.  31.5 

31.6 

15 

13  49.  8 

13  51.3 

136S.7 

3.93 

10  or  24 

35.2 

35.  3 

36.4 

36.5 

S5.-5 

Oct        1 

12  47.0 

12  48.5 

1249.9 

3.93 

11  or  26 

39.1 

39.2 

39.3 

39.4 

3».6 

19 

16 

11  62.0 

11  63.6 

11  64.9 

3.93 

12  or  26 

43.0 

43.1 

43.8 

43.3 

4B.4 

u 

HOT.     1 

10  46.1 

10  46.6 

10  48.0 

3.93 

13  or  27 

46.9 

47.0 

47.8 

47.2 

47.4 

IS 

15 

9  60.1 

9  51.6 

9  63.0 

3.64 

14  or  28 

60.8 

61.0 

61.1 

61.2 

61.3 

It 

Dec.      1 

8  47.0 

8  48.5 

8  49.9 

3.94 

89 

64.7 

64.9 

66.0 

65.2 

66.3 

14 

16 

7  61.8 

7  63.3 

7  64.7 

3.95 

30 

68.  t 

68.8 

68.9 

69.1 

60.8 

15 

81 

62.6 

62.7 

62.9 

63.0 

63.8 

It 

40  A  MANUAL  OF  LAND  SURVEYING. 

The  surveyor  should  be  careful  to  employ  Part  II, 
Table  V,  correctly.  When  the  table  is  used  in  regular 
order,  the  "^Reduction"  may  be  taken  from  Part  II 
with  the  argument  ("Argument,"  the  quantity  on 
which  another  quantity  in  a  table  depends.)  "Day  of 
the  month  "in  left  hand  column,  or,  "  Number  of  days 
elapsed  "  in  right  hand  column,  as  may  be  preferred.  In 
example  2,  Part  II,  may  be  entered  in  with  the  argu. 
ment  13  days  elapsed  (from  1st  to  14th)  in  right  hand 
column ;  then  the  reduction,  51m.l,  results,  as  above 
written;  but  when  working  from  a,  following  date  (exam- 
ple 3),  the  day  of  the  month  in  left  hand  column  cannot 
be  used. 

Mistakes  are  often  made  by  using  the  wrong  column 
in  Part  I ;  as  a  matter  of  course,  the  time  should  always 
be  taken  out  for  the  current  year. 

Applications  of  Tables  Y  and  VII. 

4.  Required  the  Hour  Angle  and  Azimuth  of  Polaris,  for  a  station  in 
latitude  46°  N.,  longitude  90°  WM  at  8h  24™  p.  m  ,  November  7,  1910. 

h.  m. 

Astronomical  time  of  observation,  1910,  Nov.  7 8  24.0 

Equivalent  to  time  of  Nov.  6 3224.0 

h.  m. 

A  stron.  time.  U.  C.  Polaris,  Nov.  1  (  Table  V,  Part  I) . .  10  46.6 
Reduction  to  Nov.  6»  (Part  II),  subtract t>19.7 


•By  reference  to  the  above  table,  the  surveyor  will  observe  that  the 
times  between  Nor.  1  and  15  are  greater  than  8h  24m  ;  consequently,  the 
culmination  for  one  day  earlier,  Nov.  6,  will  be  used  ;  sea  directions  on 
page  37  ;  also,  last  clause  of  example  3,  page  38. 

»  From  Part  II,  Table  V,  opposite  6th  day  of  month,  and  under  "  3  94m." 

°  To  subtract,  take  1  day  from  Nov.  7,  and  add  its  equivalent,  24h,  to  8* 
24m,  making,  Nov.  6,  32&  240*  (which  is  the  time  expressed  by  Nov.  7,  8"» 
24n») ;  then  subtract  in  the  usual  manner. 

dSee  last  clause  of  footnote,  page  40. 

»In  case  the  Hour  Angle  comes  out  greater  than  llh  58"»,  subtract  it  from 
23b  56. lm;  see  example  4,  on  above. 

'  The  Hour  Angle  being  less  tfian  llh  58»,  the  Azimuth  is  west;  tee  pr»- 
cepts,  top  of  Table  VII. 


TO    FIND    A   TRUE    MERIDIAN.  £j 

Astron.  time,  U.  C.  Polaris,  HOT.  6  10  26.9,  subt.  «10  28.9 


Hour  Angle  of  Polaris,  at  observation 21  57.1 

Subtract  from ...  *28  56.1 


Time  Argument  for  Table  VH 1  59.G 

Azimuth  of  Polaris,  at  observation 0°  51'  E 

5.  Required  the  Hour  Angle  and  Azimuth  of  Polaris,  for  a  station  in 
Iatitode41°  15T  N.,  longitude  94°  W.,  at  6»  16-»  a.  m.,  Nov.  19,  1901. 

h.  m- 

Astronomical  time  of  observation,  1901,  Nov.  18 18  16.O 

h.  m. 

Astron.  time,  U.  G.  Solaris,  Nov.  15  (Table  V,  Part  I),  9  46.4 
Reduction  to  Nov.  18  (Part  II)  subtract 11.8 

Aatron.  time,  U.  C.  Polaris,  Nov.  18 9  34.6,  subt.  9  34.8 

Hour  Angle  of  Polaris,  at  observation,  and  Time  Argument  for 

Table  VII ,  «8  41.4 

Azimuth  of  Polaris,  at  observation  (Table  VH),  74'  or 1°  14*  W. 

TABLE  VII. 

This  table  gives,  for  various  hour  angles,  expressed  in 
mean  solar  time,  and  for  even  degrees  of  latitude  from  30 
to  50  degrees,  the  Azimuths  of  Polaris  for  11  years,  com- 
puted for  average  values  of  the  north  polar  distance  of 
the  star — the  arguments  (reference  numbers),  being 
the  hour  angle  (or  23h  56m.l,  minus  the  hour  angle,  when 
the  latter  exceeds  llh  58m),  which  is  termed  the  Time 
Argument;  and  the  latitude  of  the  place  of  observation. 
The  table  is  so  extended  that  azimuths  may  be  taken 
out  by  mere  inspection,  and  all  interpolation  avoided, 
except  such  as  can  be  performed  mentally. 

Tbe  vertical  diameter  SS',  Plato  I,  Fig.  2,  divides  the  apparent  path  of 
Polaris  into  two  equal  parts,  and  for  the  star  at  any  point  s6  on  the  east 
side,  there  is  a  corresponding  point  slf  on  the  uxst  side  of  the  meridian, 
for  which  the  azimuth  Nw  is  equal  to  the  azimuth  Ne.  The  arc  SSj.S's,, 
taken  from  the  entire  circle  (or  23*  56».l),  leaves  the  arc  Ss6,  and  its  equal, 
Ss,,  expressed  in  time,  may  be  used  to  find,  from  Table  VII,  the  azimuth 
Nw,  which  is  equal  to  Ne. 

The  hour  angles  entered  in  Table  VII  include  only  tho»e  of  the  west  half 


42  A   MANUAL  OF  LAND  SURVEYING. 

of  the  circle  ending  at  S',  and  when  an  hour  angle  greater  than  lib  58m  re 
suits  from  observation,  it  will  be  subtracted  from  23h  56™.!,  and  the  re- 
mainder will  be  used  as  the  "  time  argument "  for  the  table.  The  surveyor 
should  not  confound  these  two  quantities.  The  hour  angle  itself  always 
decides  the  direction  of  the  azimuth  and  defines  the  place  of  the  star  with 
reference  to  the  pole  and  meridian,  as  noted  at  top  of  Table  VII.  See  ex- 
amples following  Table  V. 


The  hours  of  the  "  time  arguments  "  are  placed  in  the 
columns  headed  "Hours,"  on  left. of  each  page  The 
minutes  of  the  time  arguments  will  be  found  in  the  col- 
umns marked  "  m.,"  under  the  years  for  which  they  are 
computed,  arid  they  are  included  between  the  same 
heavy  zigzag  lines  which  inclose  the  hours  to  which 
they  belong. 

The  time  arguments  are  given  to  the  nearest  half 
minute  ;  the  occurrence  of  a  period  after  the  minutes  of 
any  one  of  them,  indicates  that  its  value  is  0.5™  greater 
than  printed,  the  table  being  so  arranged  to  economize 
space. 

The  tables  will  be  used  as  follows:  Find  the  hours  of 
the  time  argument  in  the  left-hand  column  of  either 
page  ;  then,  between  the  heavy  lines  which  inclose  the 
hours,  find  the  minutes  in  the  column  marked  at  the  top 
with  the  current  year.  On  the  same  horizontal  line 
with  the  minutes,  the  azimuth  will  be  found  under  the 
given  latitude,  which  is  marked  at  the  top  of  the  right- 
hand  half  of  each  page.  Thus,  for  1904,  time  argument, 
Oh  43m,  latitude  36 ;  find  Oh  on  left-hand  page  and  under 
1904,  find43mon  ninth  line  from  the  top,  and  on  same, 
line  with  the  minutes,  under  latitude  36,  is  the  azimuth 
0°  17.  For  1908,  time  argument.  9. 33im,  lat.  48°,  the  azi- 
muth is  1°  H',  found  on  the  21st  line  from  top  of  right- 
hand  page. 

If  the  exact  time  argument  is  not  found  in  the  table, 
the  azimuth  should  be  proportioned  to  the  difference 
between  the  given  and  tabular  values  of  said  argument- 

The  table  has  been  arranged  to  give  the  azimuths,  by 
simple  inspection.  No  written  arithmetical  work  is  re- 
quired, all  being  performed  mentally.  It  will  always  be 


TO    FIND    A    TRUE    MERIDIAN.  43 

sufficient  to  take  the  nearest  whole  degree  of  latitude, 
and  use  it  as  above  directed  except  for  a  few  values 
near  the  top  of  either  page,  where  the  difference  of 
azimuths,  for  2°  difference  of  latitude,  amounts  to  4  or  5 
minutes  of  arc. 

The  attention  of  the  surveyor  is  directed  to  the  fact 
that  he  should  always  use  one  day  of  twenty-four  hours 
as  the  unit  when  he  subtracts  the  time  of  culmination 
from  the  time  of  observation.  See  example  4,  page  40 . 
In  any  case  when  the  time  or  upper  culmination,  taken 
from  Table  V,  for  the  given  date,  would  be  numerically 
greater  than  the  astronomical  time  of  observation,  the 
former  time  will  be  taken  out  for  a  date  one  day  earlier 
than  the  date  of  observation.  The  surveyor  will  decide 
when  such  condition  exists  by  comparing  the  time  given 
in  the  table  with  his  astronomical  time  of  observation. 
See  example  4  and  explanations  following  Table  V, 
page  39. 

The  watch  time  to  be  used  when  making  observations 
on  Polaris  at  all  times  except  elongation  should  be  as 
accurate  as  can  be  obtained.  Looking  at  Table  VII, 
near  the  top  of  the  page,  the  surveyor  will  observe,  that 
for  a  difference  of  four  minutes  in  the  time  argument, 
there  is  a  change  of  about  two  minut  es  in  azimuth ;  conse- 
quently, to  obtain  the  azimuth  to  tue  nearest  whole  minute 
of  arc,  the  local  mean  time,  upon  which  all  depends,  should 
be  known  within  two  minutes.  When  the  surveyor  uses  a 
solar  instrument,  he  can  readily  determine  the  time  for 
himself  during  the  afternoon  before  observing  Polaris, 
or  in  the  morning  after  observation,  and,  without  mov- 
ing the  hands  of  his  watch,  apply  the  necessary  correc- 
tion to  his  observed  watch  time.  When  the  surveyor 
uses  standard  railroad  time,  he  will  correct  the  same  for 
the  difference  of  longitude  between  his  station  and  the 
standard  meridian  for  which  the  time  is  given,  at  the 
rate  of  four  minutes  of  time  for  each  degree  of  the  differ- 
ence in  arc.  Thus,  if  the  difference  of  longitude  is  6° 
45',  the  equivalent  in  time  will  be  27  minutes.  The 
difference  of  longitude  may  be  taken  from  a  good  .map. 


TABLE  VJL—  Azimuth  of  Polarit 

[The  hour  Angles  are  expressed  in  man  tolar  time.    The  occurrence  of  a  period  after 


STAR  AND  AZIMUTH. 
W.  of  N.  when  hour  angle  is  leu  than  llb  53=. 
E.  ot  N.  when  hour  angle  is  ffreoKr  than  11"  68». 

Time  argument,  the  star's  hour  angle  (or  23'  56"  .1 
minus  the  star's  hour  angle),  for  the  year  — 

POLARIS  abote  THI  POLL 
To  determine  the  true  meridian,  the  azimuth 
will  be  laid  off  to  the  «ut  when  the  hour  angle 
is  leu  than  11*  68-,  and  to  the  wut  when 
greater  thai)  11*  68". 

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for  the  UM  of  land  surveyors. 

miante*  of  time  or  of  an  hour  angle  Indicate*  that  its  value  U  0".5  greater  than  printed.] 


STAB  ASD  AZIMUTH. 
W.  of  X.  when  boar  angle  i*  lem  than  n>  58». 
I.  of  N.  when  hoor  angle  i*  grtaltr  tban  Ilk  *g». 

Time  argument,  the  «t*r'«  boor  angle  {or  23*  56» 
nuiuu  tbe  itar'i  boor  angle),  for  the  rear— 

J 

POLABU  MOV  TBS   POLB. 

To  determine  the  true  meridian,  the  azimuth 
will  be  laid  off  to  the  tax  when  tbe  boar 
angle  to  leo  than  Ilk  58»,  and  to  the  we*  wb«o 

?r«K*r  than  Ilk  6S-. 

i 

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53 
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52 

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67 
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62 
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42. 

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SI 
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2s. 

29 

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53 

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57. 

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67 

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'24 
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IS. 

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53 
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53 
53 

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48 
53 
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3 

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n. 

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17. 

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17. 
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17 
22 

33 

38 

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4* 

53 

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33 

38 

43 

48 
63 

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33 
3* 

43 
48 

53 

15 

33 

38 

43 
48 
53 
U 

33 

38 

43 

48 
53 

Li 

32. 
3e 

43 

4? 
63 

32. 
37 

43 

4? 
53 
Co 

6 

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r 

—  1 

r  ABLE  XII. —  Convergence/  of  Meridians  six  miles  long  ana 
miles  apart,  and  other  relevant  data,  to  latitude  70"  north. 


Lat- 
itude. 

Convergency. 

Difference  of  longi- 
tude per  range. 

Difference  of  latitude 
for— 

On  the 
parallel. 

Angle. 

In  arc. 

In  time. 

1  mile  in 
arc. 

1  Tp.  in 
arc. 

0 

Links. 

/    // 

'       U, 

Seconds. 

- 

80 

41.9 

3    0 

6    0.36 

24.02 

31 

43.6 

3.  7 

6    4.02 

24.27 

32 

45.4 

815 

6    7,93 

24.53 

0/i871 

5'.  225 

33;i 

47.2 

323 

6  12.00 

24.80 

34 

49.1 

330 

6  16.31 

25,09 

35 

50.9 

388 

620,95 

25.40 

36 

52.7 

346 

625.60 

25.71 

37 

54.7 

355 

6  30,59 

26.04 

0'.87/> 

5'.  221 

38 

56.8 

4    4 

6  35.81 

26.39 

39 

58.8 

413 

6  41.34 

26.76 

40 

60.9 

422 

6  47.13 

27.14 

41 

63.1 

4  31 

653.22 

27.55 

42 

65.4 

441 

6  59.  62 

27.97 

0'.869 

5?.  217 

43 

67.7 

451, 

7    6.27 

28.42 

44 

70.1 

5    1 

7  13.44 

28.90 

45 

72.6 

5  12 

7  20.93 

29.39 

46 

75.2 

523 

7  28.81 

29.92 

47 

77.8 

534 

7  37.10 

30.47 

0',869 

5'.  212 

48 

80.6 

546 

7  45.79 

31.05 

49 

83.5 

569 

7  55.12 

31.67 

60 

86.4 

6  12 

8    4.83 

32.32 

51 

89.6 

6  25 

*  15.  17 

33.01 

52 

92.8 

6  39 

8  26.  13 

33.74 

0'.868 

5'.  207 

53 

96.2 

6  54 

8  37.  75 

34.52 

64 

99.8 

7    9 

8  50.07 

35.34 

55 

103.5 

725 

9    3.18 

36.22 

56 

107.5 

7  42 

9  17.  12 

37.14 

67 

111.6 

8    0 

9  31.97 

38.13 

0'.867 

5',  202 

58 

116.0 

819 

9  47.83 

39.19 

59 

1216 

8  38 

10    4.78 

40.32 

60 

125.6 

859 

10  22.  94 

41.52 

61 

130.8 

9  22 

10  42.42 

42.83    . 

62 

136.3 

946 

11    3.38 

44.  22 

ova* 

6'4fl8 

63 

142.2 

10  11 

11  25.  97 

45.73 

64 

148.0 

1038 

U  50.37 

47.36 

65 

155.0 

11    8 

12  16.82 

49.12 

66 

162.8 

11  39 

12  45.55 

51.04 

67 

170.7 

12  13 

13  16.88 

53.12 

o'.see 

5U95 

68 

179.3 

12  51 

13  51.15 

55.'41 

69 

.188.7 

13  31 

1428.77 

57.90 

TO 

199.1 

14  15 

1510.26 

60,68 

P.  866 

6M98 

TO    FIND    A    TRUE    MERIDIAfc.  47 

The  number  of  seconds  taken  from  the  5th  column  of 
Table  XII  (opposite  the  proper  latitude),  multiplied 
by  the  number  of  ranges,  will  give  the  correction  for 
.longitude  in  seconds  of  time.  The  correction  will  be 
subtracted  from  the  standard  railroad  time  of  observa- 
tion, when  the  surveyor's  station  is  west,  or  added  when 
east  of  the  standard  meridian,  as  the  case  may  require, 
to  obtain  local  time.  It  is  immaterial  where  the  sur- 
veyor obtains,  the  standard  time,  provided  he  gets  it 
right,  a  result  which  will  be  determined  in  the  most 
satisfactory  manner  by  a  direct  personal  comparison 
at  a  telegraph  office. 

Table  VII  enables  the  surveyor  to  obtain  the  hour 
angle  and  azimuth  of  Polaris  at  any  hour  and  minute 
from  1901  to  1911  inclusive,  in  latitudes  30°  to  50°,  thus 
combining  in  two  pages  the  essentials  which  under 
ordinary  methods  would  require  twenty. 

Mr.  A.  W.  Barber,  expert  examiner  of  surveys  for  the 
U.  S.  Land  Department,  furnishes  the  following  method, 
which  is  accurate,  and  applicable  anywhere.  It  is  valu- 
able for  use  in  Alaska  and  other  high  latitudes,  where 
the  previous  methods  are  difficult  to  apply,  and  liable  to 
many  serious  errors:  — 

"  THE  METHOD  BY  EQUAL  ALTITUDES 

depends  on  the  fact  that  the  circumpolar  stars  describe 
invariable  circular  arcs  below  the  true  pole;  and  that 
when  a  peg  is  set  on  the  ground,  to  mark  the  course  to  a 
certain  star  when  west  of  north  at  an  altitude  of  say 
35°,  and  another  peg  is  afterward  set  to  mark  the  posi- 
tion of  that  star,  when  it  has  passed  to  the  northeast 
and  risen  to  the  same  precise  elevation,  the  meridian  of 
the  transit  will  lie  exactly  midway  between  the  pegs. 
It  may  be  used  by  any  careful  observer  having  a  good 
common  transit  with  vertical  arc. 

This  process  requires  no  reflecting  eye-piece;  no  full 
vertical  circle;  no  mathematical  tables;  no  calculations 
of  local  mean  time,  standard  time,  sidereal  time,  or  as- 
tronomical and  civil  day.  It  does  not  depend  on  the 
date  nor  the  hour  of  the  day,  nor  any  reckoning  of  cul- 
mination, hour-angle,  elongatio'n,  or  polar  azimuth. 
Neither  does  the  sun's  declination  or  the  atmospheric 
refraction  enter  at  all  into  the  calculation. 

Should  there  be  unsuspected  error  in  the  graduation 
or  setting  of  the  vertical  arc,  or  some  defect  of  collima- 


48  A    MANUAL    OF    LAND    SURVEYING, 

tion  in  the  telescope,  it  would  equally  affect  both  parts 
of  the  observer's  work,  and  produce  no  effect  in  the  re- 
sulting meridian.  Proceed  as  follows  in  any  latitude  or 
time  of  the  year  when  the  stars  are  sufficiently  discern- 
ible: — 

1.  Choose  a  suitable  station,  set  transit  firmly,  and 
level    precisely,    by   the    telescope    level    turned    in    all 
directions. 

2.  Set  the  index  arm  for  vertical  angles  at  zero,  and 
keep  it  tightly  clamped  for  reading  elevation  angles. 

3.  Provide  suitable  illumination  for  cross-wires,  and 
also  for  reading  angles,  horizontal  and  vertical. 

4.  Select  a  conspicuous  star,  perhaps  30°  or  40°  from 
the  pole,  which  appears  two  hours  more  or  less  before 
its  lower  culmination, —  that  is,  which  stands  west  of 
north  and  is  rapidly  descending.     Identify  this  star  be- 
yond all  chance  of  error,  noting  it  on  a  diagram  for  cer- 
tainty some  hours  later. 

5.  Direct  the  telescope  to  this  point,  fixing  the  star  at 
the  intersection  of  the  cross-wires,  and  clamp  the  axis 
so  it  will  retain  the  altitude  shown  on  the  arc.     Read 
and  note  down  the  angle  of  elevation,  and  read  more 
than  once. 

6.  Unclamp  the  axis,  bring  the  telescope  to  the  earth, 
and  have  an  assistant  drive  a  peg  in  line  with  the  cross- 
wire,  from  3  to  5  chains  distant.     A  candle  held  there 
before  a  white  surface,  will  exhibit  the  wire  and  give 
the  exact  point  for  a  tack. 

7.  Repeat  the  observation  once  or  twice,  at  intervals 
of  10  or  15  minutes,  for  confirmation  of  results,  marking 
successive  pegs  A,  B,  etc.,  with  degrees  and  minutes  of 
elevation  found. 

8.  Be  ready  to  observe  the  upward  path  of  the  same 
star,  after  it  has  passed  east  below  the  pole.     Correct 
the  leveling,  set  the  vertical  index  successively  at  each 
altitude  previously  noted   (beginning  with  the  lowest); 
and  when  the  star  (diagonally  ascending  in  the  field  of 
the  glass)  approaches  the  horizontal  wire,  bring  the  ver- 
tical wire  also  upon  the  star  at  the  intersection,  using 
the  slow-motion  screw  of  the  horizontal  plate. 

9.  Keep  the  plate  at  that  point,  bring  the  telescope 
down,  and  set  peg  in  line  as  before.    Repeat  the  process 
for  each  observation  A,  B,  etc.,  taken  before  midnight, 
marking   each   peg   B,   A,    etc.,   with   the    elevation    in 
figures. 

10.  Measure  the  arc  between  pegs  A  A  denoting  equal 
altitude,   and   take  one   half.     Lay   off  this   half   from 
either  peg,  and  set  a  peg  and  tack  for  the  true  meridian. 
As  a  test  of  correctness,  the  middle  point  between  pegs 


TO    FIND    A    TEUE    MERIDIAN.  49 

B  B,  and  between  C  C  should  be  found  to  coincide  with 
the  one  first  found.  A  single  pair  is  sufficient,  except 
for  confirmation. 

BY  EQUAL  ALTITUDES  OF  THE  SUN. 

In  this  operation  a  reflecting  eye-piece  with  dark 
glass  will  be  necessary.  The  sun's  large  image  in  the 
field  can  not  be  centered  as  truly  as  a  star.  It  is  there- 
fore found  best  to  place  the  intersection  of  the  wires  at 
the  lower  limb  (apparently  the  sun's  upper  edge,  as  re- 
versed by  the  mirror),  and  at  the  precise  point  of  tan- 
gency,  when  the  sun  is  just  leaving  the  horizontal  wire, 
apparently  descending,  with  the  vertical  wire  bisecting 
its  disk. 

This  is  convenient  for  the  forenoon  observations, 
hence  at  the  corresponding  times  after  noon,  with  proper 
altitude  of  telescope,  one  must  be  ready  at  the  moment 
the  sun  (now  apparently  ascending)  first  reaches  the 
level  wire,  having  the  vertical  one  bisect  the  sun  by  the 
point  of  tangency,  as  before. 

The  pair  of  pegs  in  this  case  will  be  set  southeast  and 
southwest  of  the  station.  The  center  of  bisecting  line 
of  the  included  arc  would  be  the  meridian,  were  it  not 
for  the  sun's  change  of  declination  in  the  intervening 
time.  This  slight  change  requires  a  calculation  for  cor- 
rection, which  the  star  process  avoids. 

WORKING  BY  A  REFERENCE  MARK. 

Instead  of  using  pegs  and  tacks  for  day-work  it  is 
easier  to  use  a  reference  point  or  mark. 

On  April  17,  after  careful  leveling,  I  set  the  hori- 
zontal plates  at  zero  with  the  telescope  directed  at  a 
distant  spire  for  my  mark.  Clamp  the  lower  plate  fast, 
and  direct  the  telescope  to  the  sun,  observing  it  as  above 
shown.  Find  by  the  horizontal  angle  that  the  sun's  azi- 
muth to  the  left  or  east  of  the  spire  is  30°  27';  and  by 
the  vertical  arc  I  find  his  altitude  47°  09'.  (The  semi- 
diameter  may  be  disregarded  in  each  pair  of  observa- 
tions, if  the  same  limb  of  the  sun  is  used  each  time.) 

For  the  corresponding  afternoon  observation,  I  have 
the  index  of  altitude  fixed  at  47°  09',  and  watch  the  sun 
rise  (apparently)  to  the  proper  position.  At  the  right 
moment,  clamp  the  plate,  use  the  slow-motion,  and  when 
the  disk  is  in  position,  find  from  the  horizontal  plate 
that  the  sun  is  84°  49'  west  of  the  spire  or  mark. 

The  whole  arc  is  30°  27'  4-  84°  49'  =  115°  16';  and 
the  bisecting  meridian  is  57°  38'  from  either  position  of 
the  sun.  From  this  one-half,  I  subtract  the  first  azimuth 
of  the  sun  from  the  mark;  57°  38'  —  30°  27'  =  27°  11'  as 


60  A    MANUAL    OF    LAND    SURVEYING. 

the  true  bearing  of  the  mark  from  the  uncorrected 
meridian,  and  it  apparently  bears  S.  27°  11'  E.  from  the 
transit. 

THE  CORRECTION  FOR   DECLINATION 

at  or  near  the  times  of  the  solstices,  will  be  merely 
theoretical,  as  an  hourly  difference  of  declination  less 
than  10  seconds  will  be  quite  negligible.  '  But  during  the 
rest  of  the  year  it  should  be  ascertained;  for  it  would 
amount  to  as  much  as  a  change  of  10',  were  observations 
taken  six  hours  apart  on  September  25,  in  latitude  65°. 

To  calculate  this  correction:  Take  one  half  the 
change  in  declination  between  observations  at  equal  alti- 
tude; divide  these  minutes  of  change  by  the  product  of 
the  cosine  of  the  latitude  by  the  sine  of  half  the  differ- 
ence in  time  expressed  in  degrees  (15°  per  hour);  the 
quotient  will  be  the  minutes  of  arc  for  the  correction. 
This  is  to  be  applied  from  south  to  west  from  June  21  to 
December  21  (declination  decreasing)  and  from  south  to 
east  the  rest  of  the  year. 

EXAMPLE. —  On  April  17,  in  latitude  39°,  using  stand- 
ard watch  time;  second  and  third  pairs  of  observations, 
the  first  pair  being  already  noted. 

B.  10:  01  A.  M.  Altitude  49°  36'  Azimuth  E.  from  mark 

26°  07' 

C.  10:  16  A.  M.  Altitude  51°  48'  Azimuth  E.  from  mark 

21°  38' 
C.     1:  59  P.  M.  Altitude  51°  48'  Azimuth  W.  from  mark 

76°  00' 
B.     2:  14  P.  M.  Altitude  49°  36'  Azimuth  W.  from  mark 

80°  29' 

The  sum  of  the  measured  arcs  of  the  B  B  positions, 
26°  07'  +  80°  29'  =  106°  36'.  The  middle  point  for  me- 
ridian is  at  53°  18'.  As  the  reference  mark  is  26°  07' 
from  the  forenoon  sun,  its  arc  from  the  south  meridian 
point  must  be  53°  18'  — 26°  07',  and  its  course  from  the 
transit  is  again  found  S.  27°  11'  E.  (Uncorrected.) 

The  third  pair,  C  C,  gives  21°  38' +  76°  00' =  97°  38'. 
One  half  of  this,  or  48°  49',  less  21°  38',  gives  the  same 
resulting  arc,  27°  11'. 

CORRECTION  FOR  CHANGE  OF  DECLINATION 
One  half  of  the  change  on  that  day  in  4%  hours  was 
1.9'.  The  cosine  of  the  latitude  39°  is  .78;  the  sine  of 
half  the  difference  in  time  (2y8  hours  =  about  32°)  is 
.53;  their  product  is  .78  X  .53  =  .41.  Dividing  1.9'  by  .41, 
the  quotient  is  4y2  minutes  of  arc,  for  correction  of  the 
south  meridian  point  eastward.  This  gives  the  bearing 
of  the  spire  or  mark  from  the  true  meridian,  S.  27°  06' 
30"  E. 


THE    TRANSIT. 


51 


FIG  6. 


52 


A  MANUAL  OF  LAND   SURVEYING. 


VII.  The  essential  parts  of  the 
Transit,  as  shown  in  the  cut, 
are  the  telescope  with  its  axis 
and  two  supports,  the  circular 
plates  with  their  attachments, 
the  sockets  upon  which  the 
plates  revolve,  the  leveling 
head,  and  the  tripod  on  which 
the  whole  instrument  stands. 

The  telescope  is  from  ten  to 
eleven  inches  long,  firmly  se- 
cured to  an  axis  having  its 
bearings  nicely  fitted  in  the 
standards,  and  thus  enabling 
the  telescope  to  be  moved  in 
either  direction,  or  turned  com- 
pletely around  if  desired. 

The  different  parts  of  the 
telescope  are  shown  in  Tig.  7. 

The  object-glass,  composed 
of  two  lenses,  so  as  to  show 
objects  without  color  or  dis- 
tortion, is  placed  at  the  end  of 
a  slide  having  two  bearings, 
one  at  the  end  of  the  outer 
tube,  the  other  in  the  ring  CC, 
suspended  within  the  tube  by 
four  screws,  only  two  of  which 
are  shown  in  the  cut. 

The  object-glass  is  carried 
out  or  in  by  a  pinion  working 
in  a  rack  attached  to  the  slide, 
and  thus  adjusted  to  objects 
either  near  or  remote  as  de- 
sired. 

The  eye-piece  is  made  up 
of  four  piano  convex  lenses, 
which,  beginning  at  the  eye- 
end,  are  called  respectively  the 


THE  TRANSIT, 


53 


eye,  the  field,  the  amplifying,  and  the  object  lenses,  the 
whole  forming  a  compound  microscope  having  its  focus 
in  the  plane  of  the  cross-wire  ring  BB. 

The  eye-piece  is  brought  to  its  proper  focus  usually  by 
turning  its  milled  end,  the  spiral  movement  within 
carrying  the  eye-tube  out  or  in  as  desired;  sometimes  a 
pinion,  like  that  which  focuses  the  object-glass,  is  em- 
ployed for  the  same  purpose. 

1.  The  Cross -Wires, 
(Fig.  8  ),  are  two  fibres  of 
spider-web  or  very  fine  plat- 
inum wire,  cemented  into 
the  cuts  on  the  surface  of 
a  metal  ring,  at  right  angles 
to  each  other,  so  as  to  divide 
the  open  space  in  the  center 
into  quadrants. 

2.  Optical  Axis.— The 
intersection   of    the   wires  FIG.  8 

forms  a  very  minute  point,  which,  when  they  are  adjusted, 
determines"  the  optical  axis  of  the  telescope,  and  enables 
the  surveyor  to  fix  it  upon  an  object  with  the  greatest 
precision. 

The  imaginary  line  passing  through  the  optical  axis  of 
the  telescope,  is  termed  the  Line  of  Collimation,  and  the 
operation  of  bringing  the  intersection  of  the  wires  into 
the  optical  axis  is  called  Adjusting:  the  Line  of  Ool- 
limation.  This  will  be  hereafter  described. 

3.  The  Vertical  Circle  firmly  secured  to  the  axis  of 
the  telescope  is  4J  inches  diameter,  plated  with  silver, 
divided  to  half  degrees,  and  with  its  vernier  enables  the 
surveyor  to  obtain  vertical  angles  to  single  minutes. 

4.  The  Level  on  Telescope  consists  of  a  brass  tube 
about  6}  inches  long,  each  end  of  which  is  held  between 
two  capstan-nuts  connected  with  a  screw  or  stem  attached 
to  the  under  side  of  the  telescope  tube. 


54  A   MANUAL   OF   LAND   SURVEYING. 

5.  The  Magnetic  Needle  is  four  to  five  inches  long 
in  the  different  sizes  of  transits,  its  brass  cup  having  in- 
serted in  it  a  little  socket  or  center  of  hardened  steel, 
perfectly  polished,  and  this  resting  upon  the  hardened 
and  polished  point  of  the  center-pin,  allows  the  needle 
to  play  freely  in  a  horizontal  direction,  and  thus  take  its 
direction  in  the  magnetic  meridian.    The  needle  has  its 
north  end  designated  by  a  scallop  or  other  mark,  and  on 
its  south  end  has  a  coil  of  fine  brass  wire,  easily  moved, 
so  as  to  bring  both  ends  of  the  needle  to  the  same  level. 
The  needle  is  lifted  from  the  pin  by  a  concealed  spring 
underneath  the  upper  plate,  actuated  by  a  screw  shown 
above,  thus  raising  the  button  so  as  to  check  the  vibra- 
tions of  the  needle,  or  bring  it  up  against  the  glass  when 
not  in  use,  to  avoid  the  unnecessary  wear  of  the  pivot. 

6.  The  Lower  Plate,  called  the  Limb,  is  divided  on 
its  upper  surface — usually  into  degrees  and  half -degrees — 
and  figured  in  two  rows,  Viz.,  from  0  to  360,  and  from  0  to 
90  each  way;  sometimes  but  a  single  series  is  used,  and 
then  the  figures  run  from  0  to  360  or  from  0  to  180  on 
each  side. 

7.  The  Verniers,  of  which  there  are  two  placed  op- 
posite each  other  against  the  limb,  are  auxiliary  scales 
used  in  measuring  smaller  portions  of  the  limb  than  are 
shown  by  its  graduations.    Thirty  divisions  on  the  ver- 
nier correspond  precisely  with  twenty-nine  half  degrees 
on  the  limb.    Hence  one  division  on  the  limb  exceeds  one 
division  on  the  vernier  by  one-thirtieth  of  one-half  of  a 
a  degree,  that  is,  by  one  minute. 

Accordingly,  the  number  of  any  division  of  the  vernier, 
on  the  side  toward  which  the  vernier  is  moved,  which  co- 
incides with  a  division  of  the  limb  is  tho  number  of 
minutes  of  arc  intercepted  by  the  zero  of  the  vernier  and 
the  last  preceding  division  of  the  limb. 

Thus,  by  the  device  of  a  vernier  we  are  enabled  to 
measure  angles  to  within  one  minute  although  the  limb 
of  the  transit  is  graduated  only  to  half-degrees. 


THE  TRANSIT.  55 

Adjustments.— The  principal  adjustments  of  the  Tran- 
sit are — 

(1)  The  Levels. 

(2)  The  Line  of  Collimation. 

(3)  The  Standards. 

8.  To  Adjust  the  Levels.— Set  up  the  instrument 
upon  its  tripod  as  nearly  level  as  may  be,  and  having  un- 
damped the  plates,  bring  the  two  levels  above  and  on  a 
line  with  the  two  pairs  of  leveling  screws;  then  with  the 
thumb  and  first  finger  of  each  hand  clasp  the  heads  of 
two  opposite,  and,  turning  both  thumbs  in  or  out,  as 
may  be  needed,  bring  the  bubble  of  the  level  directly 
over  the  screws,  exactly  to  the  centre  of  the  opening. 
Without  moving  the  instrument  proceed  in  the  same 
manner  to  bring  the  other  bubble  to  its  centre;  after 
doing  this,  the  level  first  corrected  may  be  thrown  a  little 
out;  bring  it  in  again;  and  when  both  are  in  place,  turn 
the  instrument  half-way  around;  if  the  bubbles  both 
come  to  the  centre,  they  would  need  no  correction,  but  if 
not,  with  the  adjusting  pin  turn  the  small  screws  at  the 
end  of  the  levels  until  the  bubbles  are  moved  over  half 
the  error;  then  bring  the  bubbles  again  into  the  centre  by 
the  leveling  screws,  and  repeat  the  operation  until  the 
bubbles  will  remain  in  the  center  during  a  complete  rev- 
olution of  the  instrument,  and  the  adjustment  will  be 
correct. 

9.  To  Adjust  the  Line  of  Collimation.— To  make 
this  adjustment— which  is,  in  other  words,  to  bring  the 
intersection  of  the  wires  into  the  optical  axis  of  the  tel- 
escope, so  that  the  Instrument,  when  placed  in  the  middle 
of  a  straight  line,  will,  by  the  revolution  o'f  the  telescope, 
cut  its  extremities— proceed  as  follows: 

Set  the  instrument  firmly  on  the  ground  and  level  it 
carefully;  und  then  having  brought  the  wires  into  the 
focus  of  the  eye-piece,  adjust  the  object-glass  on  some 
well-defined  point,  as  the  edge  of  a  chimney  or  other 
object,  at  a  distance  of  from  two  hundred  to  five  hundred 


56  A  MANUAL  OF  LAND  SURVEYING. 

feet;  determine  if  the  vertical  wire  is  plumb,  by  clamping 
the  instrument  firmly  and  applying  the  wire  to  the  verti- 
cal edge  of  a  building,  or  observing  if  it  will  move  par- 
allel to  a  point  taken  a  little  to  one  side;  should  any  dev- 
iation be  manifested,  loosen  the  cross-wire  screws,  and  by 
the  pressure  of  hand  on  the  head  outside  the  tube,  move 
the  ring  around  until  the  error  is  corrected. 

The  wires  being  thus  made  respectively  horizontal  and 
vertical,  fix  their  point  of  intersection  on  the  object 
selected;  clamp  the  instrument  to  the  spindle,  and  having 
revolved  the  telescope,  find  or  place  some  good  object  in 
the  opposite  direction,  and  at  about  the  same  distance 
from  the  instrument  as  the  first  object  assumed. 

Great  care  should  always  be  taken  in  turning  the  teles- 
cope, that  the  position  of  the  instrument  upon  the  spindle 
is  not  in  the  slightest  degree  disturbed. 

Now,  having  found  or  placed  an  object  which  the  ver- 
tical wire  bisects,  unclamp  the  instrument,  turn  it  half 
way  around,  and  direct  the  telescope  to  the  first  object 
selected;  having  bisected  this  with  the  wires,  again  clamp 
the  instrument,  revolve  the  telescope,  and  note  if  the  ver- 
tical wire  bisects  the  second  object  observed. 

Should  this  happen,  it  will  indicate  that  the  wires  are 
in  adjustment,  and  the  points  bisected  are  with  that  of 
the  centre  of  the  instrument,  in  the  same  straight  Iine0 

If  not,  however,  the  space  which  separate  the  wires 
from  the  second  point  observed^will  be  double  the  devia- 
tion of  that  point  from  a  true  straight  line,  which  may 
be  conceived  as  drawn  through  the  first  point  and  the 
centre  of  the  instrument,  since  the  error  is  the  result  of 


C 

-B 
FIG.  9. 

two  observations,  made  with  the  wires  when  they  are  out 
of  the  optical  axis  of  the  telescope. 


THE  TRANSIT.  57 

For,  as  in  the  diagram,  let  A  represent  the  centre  of  the 
instrument,  and  BC  the  imaginary  straight  line,  upon  the 
extremities  of  which  the  line  of  collimation  is  to  be  ad- 
justed. 

B  represents  the  object  first  selected,  and  D  the  point 
which  the  wires  bisected,  when  the  telescope  was  made 
to  revolve. 

When  the  instrument  is  turned  half  around,  and  the 
telescope  again  directed  to  S,  and  once  more  revolved,  the 
wires  will  bisect  an  object,  E,  situated  as  far  to  one  side 
of  the  true  line  as  the  point  D  is  on  the  other  side. 

The  space,  DE,  is  therefore  the  sum  of  two  deviations 
of  the  wires  from  a  true  straight  line,  and  the  error  is 
made  very  apparent. 

In  order  to  correct  it,  use  the  two  capstan  head  screws 
on  the  sides  of  the  telescope,  these  being  the  ones  which 
affect  the  position  of  the  vertical  wire. 

Remember  that  the  eye-piece  inverts  the  position  of 
the  wires,  and  therefore  that  in  loosening  one  of  the 
screws  and  tightening  the  other  on  the  opposite  side,  the 
operator  must  proceed  as^if  to  increase  the  error  observed. 
Having  in  this  manner  moved  back  the  vertical  wire 
until,  by  estimation,  one-quarter  of  the  space,  DE,  has 
been  passed  over,  return  the  instrument  to  the  point  B, 
i  evolve  the  telescope,  and  if  the  correction  has.  been  care- 
luily  made,  the  wires  will  now  bisect  a  point,  C  situated 
midway  between  D  and  E,  and  in  the  prolongation  of  the 
imaginary  line,  passing  through  the  point  B  and  the  cen- 
tre of  the  instrument. 

To  ascertain  if  such  is  the  case,  turn  the  instrument 
half  around,  fix  the  telescope  upon  B,  clamp  to  the  spin- 
dle, and  again  revolve  the  telescope  toward  C.  If  the 
wires  again  bisect  it,  it  will  prove  that  they  are  in  adjust- 
ment, and  that  the  points,  B,  A,  C,  all  lie  in  the  same 
straight  line. 

Should  the  vertical  wire  strike  to  one  side  of  C,  the 
error  must  be  corrected  precisely  as  above  described,  until 
it  is  entirelv  removed. 


58  A  MANUAL  OF  LAND  SURVEYING. 

10.  To  Adjust  the  Standards.— In  order  that  the 
wires'  may  trace  a  vertical  line  as  the  telescope  is  moved 
up  or  down,  it  is  necessary  that  both  the  standards  of  the 
telescope  should  be  of  precisely  the  same  height. 

To  ascertain  this  and  make  the  correction,  if  needed, 
proceed  as  follows: 

Having-  the  line  of  collimation  previously  adjusted,  set 
up  the  instrument  in  a  position  where  points  of  observa- 
tion, such  as  the  point  and  base  of  a  lofty  spire,  can  be 
selected,  giving  a  long  range  in  a  vertical  direction. 

Level  the  instrument,  fix  the  wires  on  the  top  of  the 
object  and  clamp  to  the  spindle;  then  bring  the  telescope 
down,  until  the  wires  bisect  some  good  point,  either  found 
or  marked  at  the  base;  turn  the  instrument  half  around, 
fix  the  wires  on  the  lower  point,  clamp  to  the  spindle,  and 
raise  the  telescope  to  the  highest  object. 

If  the  wires  bisect  it,  the  vertical  adjustment  is  effected; 
if  they  are  thrown  to  either  side  this  would  prove  that 
the  standard  opposite  that  side  was  the  highest,  the  ap- 
parent error  being  double  that  actually  due  to  this  cause. 

To  correct  it,  one  of  the  bearings  of  the  axis  is  made 
movable,  so  that  by  turning  a  screw  underneath  the  slid- 
ing-piece,  as  well  as  the  screws  which  hold  on  the  cap  of 
the  standard,  the  adjustment  is  made  with  the  utmost 
precision. 

11.  To  Adjust  the  Vertical  Circle. — Having  the  in- 
strument firmly  set  up  and  carefully  leveled,  bring  into 
line  the  zeros  of  the  circle  and  vernier,  and  with  the  tel- 
escope find  or  place  some  well-defined  point  or  line,  from 
one  hundred  to  five  hundred  feet  distant,  which  is  cut  by 
the  horizontal  wire. 

Turn  the  instrument  half  way  around,  revolve  the  tel- 
escope, and  fixing  the  wire  upon  the  same  point  as  before, 
note  if  the  zeros  are  again  in  line. 

If  not,  loosen  the  capstan-head  screws  which  fasten  the 
vernier,  and  move  the  zero  of  the  vernier  over  half  the 
error;  bring  the  zeros  again  into  coincidence,  and  proceed 


THE  TRANSIT.  5r 

precisely  as  at  first,  until  the  error  is  entirely  corrected 
when  the  adjustment  will  be  complete. 

It  is  not  always  convenient  to  make  this  adjustment  so 
as  entirely  to  eliminate  the  index  error.  In  this  case,  the 
error  should  be  noted  and  the  proper  correction  made  in 
measuring  a  vertical  angle. 

To  find  the  index  error  we  have  the  following 

RULE,  —  Level  the  instrument  and  direct  the  telescope 
upon  some  well  defined  spot.  Note  the  reading  of  the 
circle. 

Reverse  the  telescope  and  turn  the  vernier  plate  180°. 
Direct  the  telescope  upon  the  point  and  note  the  reading 
of  the  circle. 

Subtract  the  first  reading  from  the  second,  ami  divide 
the  remainder  by  2. 

12.    To  Run  a  Line  with  the  Transit, 

1.  Setting  up  the  Transit. — Set  the  instrument  up  over 
the  starting  point,  centreing  it  by  means  of  the  plumb 
line.  While  doing  so,  place  it  as  nearly  level  as  possible, 
leaving  as  little  as  may  be,  to  be  done  in  leveling  up  the 
plates  by  the  leveling  screws.  There  is  opportunity  for 
the  display  of  a  good  deal  of  skill  in  setting  up  a  transit 
over  a  point,  quickly,  and  in  proper  position.  For  hill 
sides,  a  tripod  having  adjustable  legs,  called  an  extension 
tripod,  is  a  great  convenience.  When  the  legs  are  not 
adjustable,  set  one  leg  of  the  tripod  down  hill  and  two 
legs  on  the  upper  side  of  the  line.  It  is  important  that 
the  instrument  should  stand  firmly  on  the  ground.  Some 
soils  are  so  yielding  that  it  is  impossible  for  the  man  at 
the  transit  to  change  the  weight  of  his  body  from  one 
foot  to  the  other,  without  getting  the  transit  out  of  posi- 
tion. One  remedy  is,  to  not  change  the  centre  of  grav- 
ity of  the  person,  after  the  transit  is  in  position,  until 
the  observation  is  taken.  Another  is,  to  drive  stout 
stakes  into  the  ground,  to  set  the  transit  legs  on.  An- 
other is  to  make  a  bridge  of  planks  or  poles' for  the  transit- 
man  to  stand  on,  so  as  to  carry  the  bearing  of  his  weight 


GO  A  MANUAL  OF   LAND  SURVEYING. 

as  far  as  possible  away  from  the  instrument.  Sometimes 
the  aid  of  an  assistant  will  need  to  be  called  in,  so  that 
the  transitman  need  not  move  around  the  transit  before 
sighting. 

When  the  transit  is  set  up  firmly  in  place,  loosen  the 
lower  clamp  and  turn  the  instrument  on  the  spindle  till 
the  level  tubes  are  each  parallel  to  an  opposite  paw  of 
the  leveling  screws. 

Turn  the  parallel  pair  of  screws  both  inward  or  out- 
ward until  the  bubble  comes  to  the  centre.  Each  level 
being  treated  in  this  way,  the  limb  of  the  instrument  Is 
caused  to  be  parallel  to  the  horizon. 

Unclamp  the  vernier  plate  and  set  the  zero  of  the  ver- 
nier to -coincide  with  the  zero  of  the  limb.  Clamp  the 
plates  in  this  adjustment.  The  leveling  screws  should  be 
kept  bearing  equally  against  the  plates. 

Do  not  turn  the  leveling  screws  up  too  tightly.  It 
tends  to  spring  the  plate  and  causes  unnecessary  wear  of 
the  screw  threads.  Simply  bring  them  to  a  firm  bearing. 

2.  Assistants  and  their  Duties.  The  Rod- 
man.—A  rodman,  often  called  a  flagman,  using  a  rod 
called  a  color  pole,  and  one  or  more  axemen  are  needed. 
The  color  pole  is  often  carried  by  the  head  chainman. 

The  man  who  carries  the  color  pole,  selects  places  to 
set  up  the  instrument,  and  gets  the  transit  points,  is  a 
very  important  factor  in  running  a  line.  Nearly  as  much 
depends  upon  him  for  accuracy  and  speed  as  upon  the 
transitman.  He  should  be  thoroughly  drilled  in  his  duty. 
He  should  hold  the  color  pole  perpendicularly,  clasping  it 
lightly  between  the  thumb  and  forefinger  of  both  hands, 
and  the  hands  held  above  the  head.  The  point  should  be 
lifted  a  little  above  the  ground  or  hub.  He  must  keep  it 
squarely  in  front  of  him,  and  move  his  body  the  same* 
distance  that  he  does  the  color  pole,  when  getting  a  point. 
As  soon  as  the  "  All  Right"  signal  is  given,  let  go  of  the 
pole.  It  will  fall  vertically  and  make  the  point  plain. 
U  the  pole  is  held  to  one  side  it  is  apt  to  nave  some 


THE  TRANSIT.  61 

uneven  pressure  given  which  will  make  it  incline  more 
or  less. 

A  man  cannot  stand  awkwardly  and  hold  a  color  pole 
accurately.  He  must  be  able  to  judge  of  the  stability  of 
the  ground  to  set  up  on.  He  must  select  places  where 
the  longest  sights  can  be  had,  and  in  running  through 
timbered  country  he  should  select  transit  points  where 
the  ground  begins  to  ascend  or  descend.  If  any  deep 
ravines  or  gullies  are  to  be  crossed,  he  must  select  points- 
to  get  across  them  with  the  least  possible  chopping,  and 
without  having  to  set  up  on  a  steep  hillside.  He  should 
not  select  a  point  on  the  shaded  side  of  a  big  tree,  but 
where  the  most  light  comes  in  through  the  leaves.  A 
small  limb  cut  out  of  the  way  will  often  let  in  a  wonder- 
ful amount  of  light,  or  a  white  handkerchief  spread  over 
the  chest,  or  a  light  colored  straw  hat  held  in  the  right 
position,  sometimes  reflects  enough  light  to  show  clearly 
objects  which  before  were  indistinct.  In  fact,  he  must  be 
a  man  of  gumption  and  equal  to  any  emergency.  But  he 
cannot  do  good  work  unless  he  is  provided  with  a  good 
color  pole. 

3.  The  Color  Pole. — It  should  be  made  from  a  good 
piece  of  straight  grained  timber.  White  or  Norway  pine 
is  good.  It  is  fitted  at  the  bottom  with  a  shoe  made  from 
gas  pipe,  with  a  steel  point  welded  on,  and  finished  by 
turning  down  in  a  machine.  The  shoe  ought  to  be  of 
sufficient  weight  to  bring  the  centre  of  gravity  within 
two  feet  of  the  bottom,  so  that  it  will  have  a  greater 
tendency  to  hang  vertically  when  held  up. 

The  sizes  of  color  poles  vary  according  to  the  places 
where  they  are  used.  If  one  is  dressed  down  with  planes 
to  a  six  or  eight-sided  stick,  tapering  slightly  toward  the 
top,  it  will  keep  straight  much  longer  than  a  stick  turned 
in  a  lathe.  The  shoe  should  be  made  of  sufficient  size  to 
receive  the  stick,  without  dressing  it  down  to  go  into  the 
socket.  When  finished  it  should  be  thoroughly  tested,  to 
see  if  the  point  of  the  shoe  has  been  set  in  line  with  the 


A   MANUAL   OF   LAND   SURVEYING. 

centre  of  the  pole.  Suspend  a  plumb  bob  from  a  point 
in  a  ceiling,  and  mark  on  the  floor  the  point  carried  down. 
Fasten  a  string  in  the  centre  of  the  top  of  the  color  pole 
and  suspend  it  from  the  same  point.  If  the  point  of  the 
shoe  covers  the  mark  on  the  floor  it  is  all  right.  Prying 
with  a  color  pole  should  be  prohibited. 

4.  Axeman.    The  axemen  provide  pickets  for  back- 
sights, clear  the  line  of  brush  and  trees,  and  drive  stakes 
and  hubs  for  transit  points.    They  should  keep  close  to 
the  line,  so  that  in  clearing  through  woods  they  do  no 
unnecessary  cutting.    A  clear  line  two  feet  wide  through 
the  brush  is  generally  all  that  is  needed.  Hubs  for  transit 
points  should  be  cut  square  on  top  and  driven  firmly  into 
the  earth,  nearly  level  with  the  surface. 

5.  Projecting  the  Line.    The  flagman  selects  the 
point  and,  facing  the  transitman,  holds  the  color  pole 
directly  in  front  of  him,  and  guided  by  the  transitman, 
places  it  in  line  and  makes  a  mark  in  the  ground.    The 
axeman  then  drives  a  hub  at  the  place  and  the.  rodman 
again  holds  up  his  pole  and  finds  the  exact  point  where 
the  line  crosses  the  hub  and  a  tack  is  driven.    Foremost 
surveys  a  line  within  the  limits  of  a  tack  head  is  consid 
ered  close  enough. 

The  hub  for  transit  point  should  not  be  driven  near  a 
large  tree,  in  soft  ground,  as  a  breeze  will  cause  the  tree 
to  sway  so  as  to  move  the  earth  for  many  feet  around  it. 
For  a  backsight  it  is  a  good  plan  to  set  up  a  picket, 
pointed  at  the  top,  so  that  the  point  shall  coincide  with 
the  hole  in  the  eye  piece  of  the  telescope.  Or  it  may  be 
set  far  enough  from  the  transit  so  that  the  point  may  be 
aligned  by  the  instrument.  The  picket  should  be  set  so 
firmly  in  the  ground  that  it  will  retain  its  place  as  long 
as  it  is  needed.  A  root  will  sometimes  so  press  against 
a  picket  as  to  throw  the  point  out  of  line  after  it  is  set. 
It  may  be  necessary  to  drive  the  picket  with  the  axe  and 
then  insert  a  wooden  point  in  a  cleft  in  the  top  of  the 
picket.  Several  such  points  set  up  in  the  line  before  the 
transit  is  moved  help  to  secure  accuracy  in  the  line. 


THE  TRANSIT.  63 

When  the  backsight  is  set,  the  transit  is  taken  forward 
and  set  up  over  the  tack  point  in  the  hub.  The  lower 
clamp  is  loosened,  the  telescope  reversed  and  sighted  to 
the  backsight  and  the  instrument  clamped  in  that  posi- 
tion. The  telescope  is  then  righted  and  the  line  contin- 
ued to  the  next  tack  point.  When  two  or  more  backsight 
points  are  visible  at  once,  any  error  in  the  adjustment  of 
the  instrument  or  in  running  the  line  will  be  readilv 
detected,  and  the  proper  correction  may  be  applied. 

If  the  line  of  collimation  is  out  of  adjustment  and  it  is . 
not  desirable  to  stop  and  adjust  it,  the  lower  clamp  is 
loosened,  the  instrument  turned  half  way  round  and 
clamped  on  the  backsight.  The  telescope  is  then  reversed 
on  its  axis  and  a  second  point  marked  beside  the  first. 
(See  Fig.  9.)  A  tack  is  thea  driven  in  the  true  line,  which 
is  midway  between  the  two.  If  the  instrument  is  much 
out  of  adjustment  it  may  be  necessary  to  drive  three 
hubs  for  this  purpose.  The  transit  is  then  set  up  in  the 
true  line,  and  the  line  continued  as  far  as  necessary,  in 
the  same  manner.  Obstacles  in  line  are  passed  by  offsets 
to  parallel  lines,  in  the  same  manner  as  when  running 
lines  by  pickets  or  compass.  Other  methods  will  be  con- 
sidered in  connection  with  Angular  Measurements. 

Examples,  to  be  solved  by  the  student  in  the  field: 

1.  Run  a  line  half  a  mile  and  mark  four  or  more 
points  along  the  line  with  hubs  and  tacks. 

2.  Retrace  it  in  the  opposite  direction,  testing  the 
points  to  see  how  they  agree. 

3.  Run  a  line  over  a  hill,  marking  points  at  the  top 
and  bottom  and  along  the  slopes. 

4.  Retrace  it  in  the  opposite  direction,  testing  the 
points. 

5.  Run  a  line  across  a  valley,  marking  points,  and  re- 
trace it  in  the  opposite  direction,  testing  tiie  points. 


64 


A  MANUAL  OF  LAND  SURVEYING. 


CHAPTEE  III. 
DESCRIPTION  OF  INSTRUMENTS,  CONTINUED. 


FIG.  10.    THE  SOLAR  COMPASS. 


THE    SOLAR    COMPASS  65 

1.  The    Solar   Compass  is  an  instrument  for 
utilizing  the  sun's  rays  to  determine  a  true  meridian 
which  has  for  a  number  of  years  been  in  use  in  the  sur- 
veys of  United  States  public  lands,  the  principal  lines 
of  which  are  required  to  be  run  with  reference  to  the 
true  meridian. 

The  arrangement  of  its  sockets  and  plates  is  similar  to 
that  of  the  surveyor's  transit,  except  that  the  sight  vares 
are  attached  to  the  under  plate  or  limb,  and  this  revolves 
around  the  upper  or  vernier  plate  on  which  the  solar  ap- 
paratus is  placed. 

The  limb  is  divided  to  half  degrees,  is  figured  in  two 
rows,  as  usual,  and  reads  by  the  two  opposite  verniers  to 
single  minutes. 

2.  The  Solar  Apparatus  is  seen  in  the  place  of 
the  needle,  and  in  fact  operates  as  its  substitute  in  the 
field. 

It  consists  mainly  of  three  arcs  of  circles,  by  which  can 
be  set  off  the  latitude  of  a  place,  the  declination  of  the 
sun,  and  the  hour  of  the  day. 

These  arcs,  designated  in  the  cut  by  the  letters  a,  6.  and 
c,  are  therefore  termed  the  latitude,  the  declination, 
and  the  hour  arcs,  respectively. 

3.  The  Latitude  Arc,  a,  has  its  centre  of  motion 
in  two  pivots,  one  of  which  is  seen  at  d,  the  other  is  con- 
cealed in  the  cut. 

It  is  moved  either  up  or  down  within  a  hollow  arc, 
seen  in  the  cut,  by  a  tangent- screw  at  f,  and  is  securely 
fastened  in  any  position  by  a  clamp-screw. 

The  latitude  arc  is  graduated  to  quarter  degrees,  and 
reads  by  its  vernier,  e,  to  single  minutes;  it  has  a  range 
of  about  thirty-five  degrees,  so  as  to  be  adjustable  to  the 
latitude  of  any  place  in  the  United  States. 

4.  The  Declination  Arc,  6,  is  also  graduated  to 
quarter  degrees,  and  has  a  range  of  about  twenty-eight 
degrees. 


60  A  MANUAL  OF  LAND   SURVEYING. 

Its  vernier,  v,  reading  to  single  minutes, "is  fixed  to  a 
movable  arm,  h,  having  its  center  of  motion  at  the  end  of 
the  declination  arc,  at  g;  the  arm  is  moved  over  the  sur- 
face of  the  declination  arc,  and  its  vernier  set  to  any 
reading  by  turning  the  head  of  the  tangent-screw,  ft.  It 
is  also  securely  clamped  in  any  position  by  a  screw,  con- 
cealed in  the  engraving. 

5.  Solar  Lenses  and  Lines.— At  each  end  of  the 
arm,  h,  is  a  rectangular  block  of  brass,  in  which  is  set  a 
small  convex  lens,  having  its  focus  on  the  surface  of  a 
little  silver  plate,  A,  (Fig.  11,)  fastened  by  screws  to  the 
inside  of  the  opposite  block. 

On  the  surface  of  the  plate  are 
marked  two  sets  of  lines,  intersect- 
ing each  other  at  right  angles;  of 
these,  bb  are  termed  the  hour  lines, 


FIG.  11.  ana  CG  the  equatorial   lines,  as 

having  reference  respectively  to  the  hour  of  the  day  and 
the  position  of  the  sun  in  relation  to  the  equator. 

In  Fig.  11  the  equatorial  lines  are  those  on  the  lower 
block,  parallel  to  the  surface  of  the  hour  arc,  c;  the  hour 
lines  are  of  course  those  at  right  angles  to  the  first. 

6.  Equatorial  Sights.— On  the  top  of  each  of  the 
rectangular  blocks  is  seen  a  little  sighting-piece,  termed 
the  equatorial  sight,  fastened  to  the  block  by  a  small 
milled  head-screw,  so  as  to  be  detached  at  pleasure. 

They  are  used,  as  will  be  explained  hereafter,  in  adjust- 
ing the  different  parts  of  the  solar  apparatus. 

7.  The  Hour  Arc,  c,  is  supported  by  the  two  pivots 
of  the  latitude  arc,  already  spoken  of,  and  is  also  connected 
with  that  arc  by  a  curved  arm,  as  shown  in  the  figure. 

The  hour  arc  has  a  range  of  about  120°,  is  divided  to 
half  degrees,  and  figured  in  two  series,  designating  both 
the  hours  and  the  degrees,  the  middle  didsion  being 
marked  12  and  90  on  either  side  of  the  graduated  lines. 

8.  The  Polar  Axis.— Through  the  center  of  the  hour 
arc  passes  a  hollow  socket,  p  containing  the  spindle  of 


THE   SOLAR   COMPASS.  67 

the  declination  arc,  by  means  of  which  this  arc  can  be 
moved  from  side  to  side  over  the  surface  of  the  hour  arc, 
or  turned  completely  round,  as  may  be  required. 

The  hour  arc  is  read  by  the  lower  edge  of  the  gradu- 
ated side  of  the  declination  arc. 

The  axis  of  the  declination  arc,  or  indeed  the  whole 
socket  p,  is  appropriately  termed  the  polar  axis. 

9.  The  Adjuster. — Besides  the  parts  shown  in  the 
cut,  there  is  also  an  arm  used  in  the  adjustment  of  the 
instrument  as  described  hereafter,  but  laid  aside  in  the 
box  when  that  is  effected. 

The  parts  above  described  constitute  properly  the  solar 
apparatus. 

Beside  these,  however,  are  seen  the  needle-box,  n,  with 
its  arc  and  tangent-screw,  t,  and  the  spirit  levels,  for 
bringing  the  whole  instrument  to  a  horizontal  position. 

10.  The  Needle  Box  has  an  arc  of  about  36°  in  ex- 
tent, divided  to  half  degrees,  and  figured  from  the  center 
or  zero  mark  on  either  side. 

The  needle,  which  is  made  as  in  other  instruments,  ex- 
cept that  the  arms  are  of  unequal  lengths,  is  raised  or 
lowered  by  a  lever  shown  in  the  cut. 

The  needle-box  is.  attached  by  a  projecting  arm  to  a 
tangent-screw,  t,  by  which  it  is  moved  about  its  center, 
and  its  needle  set  to  any  variation. 

This  variation  is  also  read  off  by  the  vernier  on  the  end 
of  the  projecting  arm,  reading  to  three  minutes  a  gradu- 
ated arc,  attached  to  the  plate  of  the  compass. 

11 .  The  Levels  seen  with  the  solar  apparatus  have 
ground  glass  vials,  and  are  adjustable  at  their  ends  like 
those  of  other  instruments. 

The  edge  of  the  circular  plate  on  which  the  solar  work 
is  placed,  is  divided  and  figured  at  intervals  of  ten  de- 
grees, and  numbered,  as  shown,  from  0  to  90  on  each  side 
of  the  line  of  sight. 


68  A  MANUAL  OF   LAND   SURVEYING. 

These  graduations  are  used  in  connection  with  a  little 
brass  pin,  seen  in  the  center  of  the  plate,  to  obtain  ap- 
proximate bearings  of  lines,  which  are  not  important 
enough  to  require  a  close  observation. 

12.  Lines  of  Refraction. — The  inside  faces  of  the 
sights  are  also  graduated  and  figured,  to  indicate  the 
amount  of  refraction  to  be  allowed  when  the  sun  is  near 
the  horizon.    These  are  not  shown  in  the  cut. 

13.  Principles  of  the  Solar  Compass.— The  inter- 
val between  two  equatorial  lines,  cc,  in  Fig.  10,  as  well 
as  between  the  hour  linos,  66,  is  just  sufficient  to  include 
the  circular  image  of  the  sun  as  formed  by  the  solar  lens 
on  the  opposite  end  of  the  revolving  arm,  7i,  Fig.  9. 

When,  therefore,  the  instrument  is  made  perfectly  hori- 
zontal, the  equatorial  lines  and  the  opposite  lenses  being 
accurately  adjusted  to  each  other  by  a  previous  operationr 
and  the  sun's  image  brought  within  the  equatorial  lines, 
his  position  in  the  heavens,  with  reference  to  the  horizon, 
will  be  defined  with  precision. 

Suppose  the  observation  to  be  made  at  the  time  of  one 
of  the  equinoxes;  the  arm  7i,  set  at  zero  on  the  declina- 
tion arc  6,  and  the  polar  axis  p,  placed  exactly  parallel  to 
the  axis  of  the  earth. 

Then  the  motion  of  the  arm  h,  if  revolved  on  the 
spindle  of  the  declination  arc*  around  the  hour  circle  c, 
will  exactly  correspond  with  the  motion  of  the  sun  in 
the  heavens,  on  the  given  day  and  at  the  place  of  obser- 
vation; so  that  if  the  sun's  image  were  brought  between 
the  lines  cc.  in  the  morning,  it  would  continue  in  the  same 
position,  passing  neither  above  nor  below  the  lines,  as 
the  arm  was  made  to  revolve  in  imitation  of  the  motion 
of  the  sun  about  the  earth. 

In  the  morning,  as  the  sun  rises  from  the  horizon,  the 
arm  h  will  be  in  a  position  nearly  at  right  angles  to  that 
shown  in  the  cut,  the  lens  being  turned  toward  the  sun, 


THE  SOLAS  COMPASS.  69 

and  the  silver  plate  on  which  his  image  is  thrown  directly 
opposite. 

As  the  sun  ascends,  the  arm  must  be  moved  around, 
until  when  h  has  reached  the  meridian,  the  graduated 
side  of  the  declination  arc  will  indicate  12  on  the  hour 
circle,  and  the  arm  h,  the  declination  arc  6,  and  the  lati- 
tude arc  a,  will  be  in.  the  same  plane. 

As  the  sun  declines  from  the  meridian,  the  arm  li  must 
be  moved  in  the  same  direction,  until  at  sunset  its  posi- 
tion will  be  the  exact  reverse  of  that  it  occupied  in  the 
morning. 

14.  Allowance  for  Declination. — Let  us  now  sup- 
pose the  observation  made  when  the  sun  has  passed  the 
equinoctial  point,  and  when  his  position  is  affected  by 
declination. 

By  referring  to  the  Almanac,  and  setting  off  on  the  arc 
his  declination  for  the  given  day  and  hour,  we  are  still 
able  to  determine  his  position  with  the  same  certainty  as 
if  he  remained  on  the  equator. 

When  the  sun's  decimation  is  south,  that  is,  from  the 
22d  of  September  to  the  20ih  of  March  in  each  year,  the 
arc  b  is  turned  toward  the  plates  of  the  compass,  as 
shown  in  the  engraving,  and  the  solar  lens,  o,  with  the 
silver  plate  opposite,  are  made  use  of  in  the  surveys. 

The  remainder  >f  the  year,  the  arc  is  turned  from  the 
plates,  and  the  other  lens  and  plate  employed. 

When  the  solar  compass  is  accurately  adjusted,  and  its 
plates  made  perfectly  horizontal,  the  latitude  of  the  place, 
and  the  declination  of  the  sun  for  the  given  day  and 
hour,  being  aho  set  off  on  the  respective  arcs,  the  image 
of  the  sun  cannot  be  brought  between  the  equatorial  lines 
until  the  polar  axis  is  placed  in  the  plane  of  the  meridian 
of  the  place,  or  in  a  position  parallel  to  the  axis  of  the 
earth.  The  slightest  deviation  from  this  position  will 
cause  the  image  to  pass  above  or  below  the  lines,  and 
thus  discover  the  error- 


70  A  MANUAL   OF  LAND   SURVEYING. 

"We  thus,  from  the  position  of  the  sun  in  the  solar  sys- 
tem, obtain  a  certain  direction  absolutely  unchangeable, 
from  which  to  run  our  lines,  and  measure  the  horizontal 
angles  required. 

This  simple  principle  is  not  only  the  basis  of  the  con- 
struction of  the  solar  compass,  but  the  sole  cause  of  its 
superiority  to  the  ordinary  or  magnetic  instrument.  For 
in  a  needle  instrument,  the  accuracy  of  the  horizontal 
angles  indicated,  and  therefore  of  all  the  observations 
made,  depends  upon  the  delicacy  of  the  needle,  and  the 
constancy  with  which  it  assumes  a  certain  direction, 
termed  the  magnetic  meridian. 

The  principal  causes  of  error  in  the  needle,  briefly 
stated,  are  the  dulling  of  the  pivot,  the  loss  of  polarity 
in  the  needle,  the  influence  of  local  attraction,  and  the 
effect  of  the  sun's  rays,  producing  the  diurnal  variation. 

From  all  these  imperfections  the  solar  instrument  is  free. 

The  sights  and  the  graduated  limb  being  adjusted  to 
the  solar  apparatus,  and  the  latitude  of  the  place  and  the 
declination  of  the  sun  also  set  off  upon  the  respective 
arcs,  we  are  able,  not  only  to  run  the  true  meridian,  or  a 
due  east  and  west  course,  but  also  to  set  off  the  horizontal 
angles  with  minuteness  and  accuracy  from  a  direction 
which  never  changes,  and  is  unaffected  by  attraction  of 
any  kind. 

15.  Adjustments. — The  adjustments  of  this  instru- 
ment, with  which  the  surveyor  will  have  to  do,  are  sim- 
ple and  few  in  number,  and  will  now  be  given  in  order. 

1st.  To  Adjust  the  Levels.— Proceed  precisely  as  di- 
rected in  the  account  of  the  other  instruments  we  have 
described,  by  bringing  the  bubbles  into  the  centre  of  the 
tubes  by  the  leveling  screws  of  the  tripod,  and  then  re- 
versing the  instrument  upon  its  spindle,  and  raising  or 
lowering  the  ends  of  the  tubes,  until  the  bubbles  will 
remain  in  the  centre  during  a  complete  revolution  of  the 
instrument. 


THE  SOLAR   COMPASS.  71 

2d.  To  Adjust  -the  Equatorial  Lines  and  Solar 
Lenses. — First  detach  the  arm  h  from  the  declination 
arc,  by  withdrawing  the  screws  shown  in  the  cut  from 
the  ends  of  the  posts  of  the  tangent-screw  Jc,  and  also 
the  clamp-screw,  and  the  conical  pivot  with  its  small 
screws  by  which  the  arm  and  declination  arc  are  con- 
nected. 

The  arm  h,  being  thus  removed,  attach  the  adjuster  in 
its  place  by  replacing  the  conical  pivot  and  screws,  and 
insert  the  clamp-screw  so  as  to*  clamp  the  adjuster  at  any 
point  on  the  declination  arc. 

Now  level  the  instrument,  place  the  arm  h  on  the  ad- 
juster, with  the  same  side  resting  against  the  surface  of 
the  declination  arc  as  before  it  was  detached.  Turn  the 
instrument  on  its  spindle  so  as  to  bring  the  solar  lens  to 
be  adjusted  in  the  direction  of  the  sun,  and  raise  or  lower 
the  adjuster  on  the  declination  arc,  until  it  can  be  clamped 
in  such  a  position  as  to  bring  the  sun's  image  as  near  as 
may  be  between  the  equatorial  lines  on  the  opposite  silver 
plate,  and  bring  the  image  precisely  into  position  by  the 
tangent  of  the  latitude  arc  or  the  leveling-screws  of  the 
tripod.  Then  carefully  turn  the  arm  half  way  over,  until 
it  rests  upon  the  adjuster  by  the  opposite  faces  of  the 
rectangular  blocks,  and  again  observe  the  position  of  the 
sun's  image. 

If  it  remains  between  the  lines  as  before,  the  lens  and 
plate  are  in  adjustment;  if  not,  loosen  the  three  screws 
which  confine  the  plate  to  the  block,  and  move  the  plate 
under  their  heads,  until  one-half  the  error  in  the  position 
of  the  sun's  image  is  removed. 

Again  bring  the  image  between  the  lines,  and  repeat 
the  operation  until  it  will  remain  in  the  same  situation, 
in  both  positions  of  the  arm,  when  the  adjustment  will 
be  completed. 

To  adjust  the  other  lens  and  plate,  reverse  the  arm,  end 
for  end,  on  the  adjuster,  and  proceed  precisely  as  in  the 
former  case,  until  the  same  result  is  attained. 


72  A  MANUAL  OP  LAND  SURVEYING. 

In  tightening  the  screws  over  the  silver  plate,  care 
must  be  taken  not  to  move  the  plate. 

This  adjustment  now  being  complete,  the  adjuster 
should  be  removed,  and  the  arm  h,  with  its  attachments, 
replaced  as  before. 

3d.  To  Adjust  the  Vernier  of  the  Declination  Aro. 

—Having  leveled  the  instrument,  and  turned  its  lens  in 
the  direction  of  the  sun,  clamp  to  the  spindle,  and  set  the 
vernier  v,  of  the  declination  arc,  at  zero,  by  means  of  the 
tangent-screw  a.  k,  and  clamp  to  the  arc. 

See  that  the  spindle  moves  easily  and  yet  truly  in  the 
socket,  or  polar  axis,  and  raise  or  lower  the  latitude  arc 
by  turning  the  tangent-screw  /,  until  the  sun's  image  is 
brought  between  the  equatorial  lines  on  one  of  the  plates. 
Clamp  the  latitude  arc  by  the  screw,  and  bring  the  image 
precisely  into  position  by  the  leveling-screws  of  the  tripod 
or  socket,  and  without  disturbing  the  instrument,  care- 
fully revolve  the  arm  h,  until  the  opposit,  lens  and  plate 
are  brought  in  the  direction  of  the  sun,  and  note  if  the 
sun's  image  comes  between  the  lines  as  before. 

If  it  does,  there  is  no  index  error  of  the  declination  arc; 
if  not,  with  the  tangent-screw  k,  move  the  arm  until  the 
sun's  image  passes  over  half  the  error;  again  bring  the 
image  between  the  lines,  and  repeat  the  operation  as 
before,  until  the  image  will  occupy  the  same  position  on 
both  plates. 

"We  shall  now  find,  however,  that  the  zero  marks  on  the 
arc  and  the  vernier  do  not  correspond,  and  to  remedy  this 
error,  the  little  flat-head  screws  above  the  vernier  must  be 
loosened  until  it  can  be  moved  so  as  to  make  the  zeros 
coincide,  when  the  operation  will  be  completed. 

4th.  To  Adjust  the  Solar  Apparatus  to  the  Compass 
Sights. — First  level  the  instrument,  and  with  the  clamp 
and  tangent-screws  set  the  main  plate  at  90°  by  the  ver- 
niers and  horizontal  limb.  Then  remove  the  clamp-screw 
and  raise  the  latitude  arc  until  the  polar  axis  is  by  esti- 


*  THE  SOLAR   COMPASS.  73 

mation  very  nearly  horizontal,  and  if  necessary,  tighten 
the  screws  on  the  pivots  of  the  arc,  so  as  to  retain  it  in 
this  position. 

Fix  the  vernier  of  the  declination  arc  at  zero,  and  direct 
the  equatorial  sights  to  some  distant  and  well  marked 
object,  and  observe  the  same  through  the  compass  sights. 
If  the  same  object  is  seen  through  both,  and  the  verniers 
read  to  90°  on  the  limb,  the  adjustment  is  complete;  if 
not,  the  correction  must  be  made  by  moving  the  sights  or 
changing  the  position  of  the  verniers. 

16.  To  Use  the  Solar  Compass.— Before  this  instru- 
ment can  be  used  at  any  given  place,  it  is  necessary  to  set 
off  upon  its  arcs  both  the  declination  of  the  sun  as  affected 
by  its  refraction  for  the  given  day  and  hour,  and  the  lat- 
itude of  the  place  where  the  observation  is  made. 

To  Set  off  the  Declination.— The  declination  of  the 
sun,  given  in  tne  ephemeris  of  the  Nautical  Almanac 
fuom  year  to  year,  is  calculated  for  apparent  noon  at 
Greenwich,  England. 

To  determine  it  for  any  other  hour  at  a  place  in  the 
United  States,  reference  must  be  had,  not  only  to  the  dif- 
ference of  time  arising  from  the  longitude,  but  also  to  the 
change  of  declination  from  day  to  day. 

The  longitude  of  the  place,  and  therefore  its  difference 
in  time,  if  not  given  directly  in  the  tables  of  the  Almanac, 
can  be  ascertained  very  nearly  by  reference  to  that  of 
other  places  given,  which  are  situated  on,  or  very  nearly 
on,  the  same  meridian. 

It  is  the  practice  of  surveyors  in  the  states  east  of  the 
Mississippi,  to  allow  a  difference  of  six  hours  for  the  dif- 
ference in  the  longitude,  calling  the  declination  given  in 
the  Almanac  for  12  M.,  that  of  6  A  M.,  at  the  place  of  ob- 
servation. 

Beyond  the  meridian  of  Santa  Fe,  the  allowance  would 
be  about  seven  hours,  and  in  California,  Oregon,  and  Wash- 
ington Territory  about  eight  hours. 


74  A  MANUAL  OF  LAND  SURVEYING. 

Having  thus  the  difference  of  time,  we  very  readily  ob- 
tain the  declination  for  a  certain  hour  in  the  morning, 
which  would  be  earlier  or  later  as  the  longitude  was 
greater  or  less,  and  the  same  as  that  of  apparent  noon  at 
Greenwich  on  the  given  day.  Thus,  suppose  the  observa- 
tion made  at  a  place,  say,  five  hours  later  than  Greenwich, 
then  the  declination  given  in  the  Almanac  for  the  given 
day  at  noon,  affected  by  the  refraction,  would  be  the 
declination  at  the  place  of  observation  for  7  o'clock  A.M.; 
this  gives  us  the  starting-point. 

To  obtain  the  declination  for  the  other  hours  of  the 
day,  take  from  the  Almanac  the  declination  for  apparent 
noon  of  the  given  day,  and,  as  the  declination  is  increas- 
ing or  decreasing,  add  to  or  subtract  from  the  decimation 
of  the  first  hour,  the  difference  for  one  hour  as  given  in 
the  ephemeris,  which  will  give,  when  affected  by  the  re- 
fraction, the  declination  for  the  succeeding  hour;  and 
proceed  thus  in  "making  a  table  of  the  declination  for 
every  hour  of  the  day. 

17.  Refraction.— By  reason  of  the  increasing  density 
of  the  atmosphere  from  its  upper  regions  to  the  earth's 
surface,  the  rays  of  light  from  the  sun  are  bent  out  of 
their  course,  so  as  to  make  his  altitude  appear  greater 
than  is  actually  the  case. 

The  amount  of  refraction  varies,  according  to  the  alti- 
tude of  the  body  observed;  being  0  when  it  is  in  the 
zenith,  about  one  minute  when  midway  from  the  horizon 
to  the  zenith,  and  almost  34'  when  in  the  horizon. 

18.  Allowance  for  Refraction. — The  proper  allow- 
ance to  be  made  for  refraction  in  setting  off  the  declina- 
tion of  the  sun  upon  the  Solar  Compass  has  long  been  a 
source  of  perplexity  to  the  surveyor.  Accordingly,  a  table 
has  been  prepared,  (Table  XI),  by  which  the  amount  of 
refraction  for  any  hour  of  the  day  throughout  the  year 
may  be  readily  obtained.    The  manner  of  using  the  table 
is  shown  in  the  solution  of  the  following 


THE  SOLAR   COMPASS.  75 

Example— \.  To  find  the  declination  for  the  different 
hours  of  April  16, 1883,  at  Troy,  N.  Y. 

Solution.— Latitude  of  Troy,  about  42°  30'  N.  Longi- 
tude, 4  hr.,  54  min.,  40  sec.,  practically  5  hr. 

Apparent  noon  at  Greenwich  is  7  A.  M.  at  Troy.  Decli- 
nation of  sun  at  Greenwich  at  noon  of  April  16, 1883,  as 
given  by  Nautical  Almanac,  K  10°  6'  2"-f,  and  hourly 
change,  53". 

Refraction  in  Lat.  42°  30',  declination  10°,  time  5  hr. 
before  noon  as  given  by  table,  1'  58". 

Whence  the  following  figures: 

N.10"   6'   2"+Ref.5hrs.  1' 58"  =10°   8'   0"  —  Dec.  at  7  A.  M.  Troy, 
add  hr.  dif.  53" 


N.  10°    6' 55"+     "    4    "     I'll"  —  10°    8'    0   .6  — 
add  hr.  dif.  53" 


N.  10°    T  48"  +     "    3    "    0'52"=-10°    8' 40"     — 
add  hr.  dif.  53" 


N.  10°    8' 41"+     "    2     "     039"— 10'    9' 20"     —      "     10 
add  hr.  dif.  53" 


N.  10°    9' 34"+     "     I     '*•   0' 36"  —  10°  10' 10"     — 
add  hr.  dif.  53" 


N.  10°  10' 27"+     "    0     "     0' 36"  —  10°  11' 03"     —        '     12  M. 
add  hr.  dif.  53" 


N.  10°  11' 20"  +     "     1     "     0' 36"  —  IT  11' 56"  IP.  M. 

add  hr.  dif.  53" 


N.  10°  12'  13"  +     **    2    "    0'  39"  —  13*  12'  52"     — 
add  hr.  dif.  53" 


N.  10°  13'  06"  +     "    3    "    0'  52"  —  10*  13'  58"     — 
add  hr.  dif  53' 


N.  10°  13' 59"+     "     4    "     1' 11  '  —  10°  15' 10"     ~ 
add  hr.  dif.  53' 


N.  10°  14' 49"  +     "     5    "     1' 58"  =  10°  16' 50"     —      '        5    " 

Example. — 2.  To  find  the  declination  for  the  different 
hours  of  Oct.  16, 1883,  at  Troy,  N.  Y. 

Solution.— Declination  of  sun  at  Greenwich  at  noon  of 
Oct.  16, 1883,  as  given  by  Nautical  Almanac  S.  8°51'47".7. 
hourly  change  55". 


76  A   MANUAL   OF   LAND    SURVEYING. 

Refraction  5  hr.  before  noon,  Lat.  42°  30',  Dec.  —  9°, is 
very  nearly  9 '  24  " ,  and  operates  to  diminish  the  declina- 
tion. 

Whence  the  following: 

S.  8*  51'  47".7— Ref.  5  hr.  9'  24"—  8"  42'  23"-=  Dec.  at  7A.M.  at  Troy. 
add  hr.  diff.  55" 


S.  8°  52' 42"    —    "     4  "    2  49"=  8°  49' 53"= 
add  hr.  diff.  55" 


8.8°  53' 37"    —    "     3"    1  49"=  8°  51' 48"— 
add  hr.  diff.  65" 


S.  8°  54' 32"    —    "    2"     1' 26"— 8°  53' 06"=        "    10 
addhr.  diff.  55" 


8.8°  55'  27"    —    "    1  "     1' 14"=  8°  54' 13"=         "    11 
add  hr.  diff.  55" 


S.  8°  56'  22"    —    "    0  "     1'  14"-=  8°  55'  08"=        "    12  M. 
addhr.  diff.  55" 


S.  8°  57' 17"    —    "     1  *•     1' 14"=  8°  56' 03"=        "      1  P.  M. 
addhr.  diff.  55" 


S.  8°  58'  12  '    —    *•    2  '•     1'  26"=  8"  56'  46"=        "      2    •* 
add  hr.  diff.  55" 

etc.  etc.  etc. 

19.  To  Set  Off  the  Latitude.— Find  the  declination 
of  the  sun  for  the  given  day  at  noon,  at  the  place  of  ob- 
servation, as  just  described,  and  with  the  tangent-screw 
set  it  off  upon  the  declination  arc,  and  clamp  the  arm 
firmly  to  the  arc. 

Observe  in  the  Almanac  the  equation  of  time  for  the 
given  day,  in  order  to  know  about  the  time  the  sun  will 
reach  the  meridian. 

Then,  about  fifteen  or  twenty  minutes  before  this  time, 
set  up  the  instrument,  level  it  carefully,  fix  the  divided 
surface  of  the  declination  arc  at  12  on  the  hour  circle,  and 
turn  the  instrument  upon  its  spindle  until  the  solar  lens 
is  brought  into  the  direction  of  the  sun. 

Loosen  the  clamp-screw  of  the  latitude  arc,  and  with 
the  tangent-screw  raise  or  lower  this  arc  until  the  image 
of  the  sun  is  brought  precisely  between  the  equatorial 
lines,  and  turn  the  instrument  from  time  to  time  so  as  to 
keep  the  image  also  between  the  hour  lines  on  the  plate 


THE  SOLAR   COMPASS.  77 

% 

As  the  sun  ascends,  its  image  will  move  below  the  lines, 
and  the  arc  must  be  moved  to  follow  it.  Continue  thus, 
keeping  it  between  the  two  sets  of  lines  until  its  image 
begins  to  pass  above  the  equatorial  lines,  which  is  also 
the  moment  of  its  passing  the  meridian. 

Now  read  off  the  vernier  of  the  arc,  and  we  have  the 
latitude  of  the  place,  which  is  always  to  be  set  off  on  the 
arc  when  the  compass  is  used  at  the  given  place. 

It  is  the  practice  of  surveyors  using  the  solar  compass 
to  set  off,  in  the  manner  just  described,  the  latitude  of 
the  point  where  the  survey  begins,  and  to  repeat  the  ob- 
servation and  correction  of  the  latitude  arc  every  day 
when  the  weather  is  favorable,  there  being  also  nearly  an 
hour  at  mid-day  when  the  sun  is  so  near  the  meridian  as 
not  to  give  the  direction  of  lines  with  the  certainty  re- 
quired. 

20.  To  Bun  Lines  with  the  Solar  Compass.— Hav- 
ing set  off  in  the  manner  just  given,  the  latitude  and 
declination  upon  their  respective  arcs,  the  instrument 
being  also  in  adjustment,  the  surveyor  is  ready  to  run 
lines  by  the  sun. 

To  do  this,  the  instrument  is  set  over  the  station  and 
carefully  leveled,  the  plates  clamped  at  zero  on  the  hori- 
zontal limb^  and  the  sights  directed  north  and  south,  the 
direction  being  given,  when  unknown,  approximately  by 
the  needle. 

The  solar  lens  is  then  turned  to  the  sun,  and  with  one 
hand  on  the  instrument,  and  the  other  on  the  revolving 
arm,  both  are  moved  from  side  to  side,  until  the  sun's 
image  is  made  to  appear  on  the  silver  plate;  when  by 
carefully  continuing  the  operation,  it  may  be  brought 
precisely  between  the  equatorial  lines. 

Allowance  being  now  made  for  refraction,  the  line  of 
sights  will  indicate  the  true  meridian;  the  observation 
may  now  be  made,  and  the  flag-man  put  in  position. 


78  A  MANUAL  OF   LAND   SURVEYING. 

When  a  due  east  and  west  line  is  to  be  run,  the  verniers 
of  the  horizontal  limb  are  set  at  90°,  and  the  sun's  image 
kept  between  the  lines  as  before. 

The  solar  compass  being  so  constructed  that  when  the 
sun's  image  is  in  position  the  limb  must  be  clamped  at  0 
in  order  to  run  a  true  meridian  line,  it  will  be  evident 
that  the  bearing  of  any  line  from  the  meridian  may  be 
read  by  the  verniers  of  the  limb  precisely  as  in  the  ordin- 
ary magnetic  compass,  the  bearings  of  lines  are  read  from 
the  ends  of  the  needle. 

21.  Use  of  the  Needle.— In  running  lines,  the  mag- 
netic needle  is  always  kept  with  the  sun ;  that  is,  the 
point  of  the  needle  is  made  to  indicate  0  on  the  arc  of  the 
compass  box,  by  turning  the  tangent-screw  connected 
with  its  arm  on  the  opposite  side  of  the  plate.    By  this 
means,  the  lines  can  be  run  by  the  needle  alone  in  case  of 
the  temporary  disappearance  of  the  sun;  but,  of  course, 
in  such  cases  the  surveyor  must  be  sure  that  no  local 
attraction  is  exerted. 

The  variation  of  the  needle,  which  is  noted  at  every 
station,  i3  read  off  in  degrees  and  minutes  on  the  arc,  by 
the  edge  of  which  the  vernier  of  the  needle-box  moves. 

22.  Allowance  for  the  Earth's  Curvature  —When 
long  lines  are  run  by  the  solar  compass,  either  by  the 
true  meridian,  or  due  east  and  west,  allowance  must  be 
made  for  the  curvature  of  the  earth. 

Thus,  in  running  north  or  south,  the  latitude  changes 
about  one  minute  for  every  distance  of  92  chains  30  links, 
and  the  side  of  a  township  requires  a  change  on  the  lati- 
tude arc  of  5 '  12 ' ' ,  the  township,  of  course,  being  six 
miles  square. 

This  allowance  is  of  constant  use  where  the  surveyor 
fails  to  get  an  observation  on  the  sun  at  noon,  and  is  a 
very  close  approximation  to  the  truth. 

In  running  due  east  and  west,  as  in  tracing  the  stand- 


THE  SOLAR  COMPASS.  70 

ard  parallels  of  latitude,  the  sights  are  set  at  90°  on  the 
limb,  and  the  line  is  run  at  right  angles  to  the  meridian. 

If  no  allowance  were  made  for  the  earth's  curvature, 
these  lines  would,  if  sufficiently  produced,  reach  the 
equator,  to  which  they  are  constantly  tending. 

Of  course,  in  running  short  lines  either  east  or  west,  the 
variation  from  the  parallel  would  be  so  small  as  to  be  of 
no  practical  importance;  but  when  long  sights  are  taken, 
the  correction  should  be  made  by  taking  fore  and  back 
sights  at  every  station,  noting  the  error  on  the  back  sight, 
and  setting  off  one-half  of  it  on  the  fore  sight  on  the  side 
toward  the  pole. 

23,  Time  of  Day  by  the  Sun.— The  time  of  day  is 
best  ascertained  by  the  solar  compass  when  the  sun  is  on 
the  meridian,  as  at  the  time  of  making  the  observation 
for  latitude. 

The  time  thus  given  is  that  of  apparent  noon,  and  can 
be  reduced  to  mean  time  by  merely  applying  the  equation 
of  time  as  directed  in  the  Almanac,  and  adding  or  sub- 
tracting as  the  sun  is  slow  or  fast. 

The  time,  of  course,  can  also  be  taken  before  or  after 
noon,  by  bringing  the  sun's  image  between  the  hour  lines, 
and  noticing  the  position  of  the  divided  edge  of  the  re- 
volving arm,  with  reference  to  the  graduations  of  the 
hour  circle,  allowing  four  minutes  of  time  for  each  de- 
gree of  the  arc,  and  thus  obtaining  apparent  time,  which 
must  be  corrected  by  the  equation  of  time  as  just  de- 
described.' 

24.  Caution  as  to  the  False  Image. — In  using  the 
compass  upon  the  sun,'if  the  revolving  arm  be  turned  a 
little  one  side  of  its  proper  position,  a  false  or  reflected 
image  of  the  sun  will  appear  on  the  silver  plate  in  nearly 
the  same  place  as  that  occupied  by  the  true  one.    It  is 
caused  by  the  reflection  of  the  true  image  from  the  sur- 
face of  the  arm,  and  is  a  fruitful  source  of  error  to  the 


80  A  MANUAL  OF  LAND  SURVEYING. 

inexperienced  surveyor.  It  can,  however,  be  readily  dis- 
tinguished from  the  real  image  by  being  much  less  bright, 
and  not  so  clearly  denned. 

25.  Approximate  Bearings.— When  the  bearings  of 
lines,  such  as  the  course  of  a  stream,  or  the  boundaries  of 
a  forest,  are  not  desired  with  the  certainty  given  by  the 
verniers  and  horizontal  limb,  a  rough  approximation  of 
the  angle  they  make  with  the  true  meridian  is  obtained 
by  the  divisions  on  the  outside  of  the  circular  plate. 

In  this  operation,  a  pencil,  or  thin  straight  edge  of  any 
sort,  is  held  perpendicularly  against  the  circular  edge  of 
the  plate,  and  moved  around  until  it  is  in  range  with  the 
eye,  the  brass  center-pin,  and  the  object  observed. 

The  bearing  of  the  line  is  then  read  off  at  the  point 
where  the  pencil  is  placed. 

Time  for  Using  the  Solar  Compass.— The  solar  com- 
pass, like  the  ordinary  instrument,  can  be  used  at  all 
seasons  of  the  year,  the  most  favorable  time  being,  of 
course,  in  the  summer,  when  the  declination  is  north,  and 
the  days  are  long,  and  more  generally  fair. 

It  is  best  not  to  take  the  sun  at  morning  and  evening, 
when  it  is  within  half  an  hour  of  the  horizon,  nor,  for 
about  the  same  interval,  before  and  after  it  passes  the 
meridian. 

II.      THE   SOLAR   ATTACHMENT. 

1.  The  Solar  Attachment  is  essentially  the  solar 
apparatus  of  Burt  placed  upon  the  cross-bar  of  the  or- 
dinary transit,  the  polar  axis  only  being  directed  above 
instead  of  below,  as  in  the  solar  compass.  A  little  circu- 
lar disk  of  an  inch  and  a  half  diameter,  and  having  a 
short  round  pivot  projecting  above  its  upper  surface,  is 
first  screwed  firmly  to  the  axis  ofthe  telescope. 

Upon  this  pivot  rests  the  enlarged  base  of  the  polar 
axis,  which  is  also  firmly  connected  with  the  disk  by  four 


THE  SOLAR  ATTACHMENT. 


81 


capstan-head  screws  passing  from  the  under  side  of  the 
disk  into  the  base  already  named. 

These  screws  serve  to  adjust  the  polar  axis,  as  will  be 
explained  hereafter. 


82  A  MANUAL   OF   LAND   SURVEYING. 

2.  The  hour  circle  surrounding  the  base  of  the  polar 
axis  is  easily  movable  about  it,  and  can  be  fastened  at  any 
point  desired  by  two  flat-head  screws  above.    It  is  divided 
to  five  minutes  of  time;  is  figured  from  I.  to  XII.,  and  is 
read  by  a  small  index  fixed  to  the  declination  circle,  and 
moving  with  it. 

A  hollow  cone,  or  socket,  fitting  closely  to  the  polar 
axis  and  made  to  move  snugly  upon  it,  or  clamped  at  any 
point  desired  by  a  milled-head  screw  on  top,  furnishes  by 
its  two  expanded  arms  below,  a  firm  support  for  the  dec- 
lination arc,  which  is  securely  fastened  to  it  by  two  large 
screws. 

3.  The  declination  arc  is  of   about   five   inches 
radius,  is  divided  to  quarter  degrees,  and  reads  by  its  ver- 
lier  to  single  minutes  of  arc,  the  divisions  of  both  vernier 
and  limb  being  in  the  same  plane. 

The  declination  arm  has  the.  usual  lenses  and  silver 
plates  on  the  two  opposite  blocks,  made  precisely  like 
those  of  the  ordinary  solar  compass,  but  its  vernier  is 
outside  the  block,  and  more  easily  read. 

The  declination  arm  has  also  a  clamp  and  tangent 
movement,  as  shown  in  the  cut.  The  arc  of  the  declina- 
tion limb  is  turned  on  its  axis  and  one  or  the  other- 
solar  lens  used,  as  the  sun  is  north  or  south  of  the 
equator. 

4.  The  latitude  is  set  off  by  means  of  a  large  verti- 
cal limb  having  a  radius  of  two  and  a  half  inches;  the 
arc  is  divided  to  thirty  minutes,  is  figured  from  the  centre, 
each  way,  in  two  rows,  viz.  from  0  to  80°,  and  from  90°  to 
10°,  the  first  series  being  intended  for  reading  vertical 
angles;  the  last  series  for  setting  off  the  latitude,  and  is 
read  by  its  vernier  to  single  minutes. 

It  has  also  a  clamp-screw  Inserted  near  its  centre,  by 
which  it  can  be  set  fast  to  the  telescope  axis  in  any  de- 
sired position. 

The  vernier  of  the  vertical  limb  is  made  movable  by 
the  tangent-screw  attached,  so  that  its  zero  and  that  oi 


THE   SOLAR   ATTACHMENT.  83 

the  limb  are  readily  made  to  coincide  when,  in  adjusting 
the  limb  to  the  level  of  the  telescope,  the  arc  is  clamped 
to  the  axis. 

The  usual  tangent  movement  to  the  telescope  axis 
serves,  of  course,  to  bring  the  vertical  limb  to  the  proper 
elevation,  as  hereafter  described. 

A  level  on  the  under  side  of  the  telescope,  with  ground 
vial  and  scale,  is  indispensable  in  the  use  of  the  Solar 
attachment. 

The  divided  arcs,  vernier,  and  hour  circle  are  all  on 
silver  plate,  and  are  thus  easily  read  and  preserved  from 
tarnishing. 

5.  Adjustments.— These  pertain  to  the  solar  lenses 
and  lines,  the  declination  arc,  the  polar  axis  and  hour  arc, 
as  follows: 

(1)  The  solar  lenses  and  lines  are  adjusted  precisely 
like  those  of  the  ordinary  Solar,  the  decimation  arm  being 
first  detached  by  removing  the  clamp  and  tangent-screws, 
and  the  conical  centre  with  its  two  small  screws,  by  which 
the  arm  is  attached  to  the  arc. 

The  adjuster,  which  is  a  short  bar  furnished  with  every 
instrument,  is  then  substituted  for  the  declination  arm, 
the  conical  centre  screwed  into  its  place,  at  one  end,  and 
the  clamp-screw  into  the  other,  being  inserted  through 
the  hole  left  by  the  removal  of  the  tangent-screw,  thus 
securing  the  adjuster  firmly  to  the  arc. 

The  arm  is  then  turned  to  the  sun,  as  described  in  the 
article  on  the  Solar  Compass,  and  reversed  by  the  opposite. 
faces  of  the  blocks  upon  the  adjuster,  until  the  image 
will  remain  in  the  centre  of  the  equatorial  lines. 

(2)  The  vernier  of  tha  declination  arc  is  adjusted 
by  setting  the  vernier  at  zero,  and  then  raising  or  lower- 
ing the  telescope  by  the  tangent-screw  until  the  sun's 
image  appears  exactly  between  the  equatorial  lines. 

Having  the  telescope  axis  clamped  firmly,  carefully 
revolve  the  arm  until  the  image  appears  on  the  other 
plate. 


84  A  MANUAL  OF  LAND  SURVEYING,, 

Jf  precisely  between  the  lines,  the  adjustment  is  com- 
plete; if  not,  move  the  declination  ,arm  by  its  tangent- 
screw,  until  the  image  will  come  precisely  between  the 
lines  on  the  two  opposite  plates;  clamp  the  arm  and  re- 
move the  index  error  by  loosening  two  screws  that  fasten 
the  vernier;  place  the  zeros  of  the  vernier  and  limb  in 
exact  coincidence,  tighten  the  screws,  and  the  adjustment 
is  finished. 

(3)  To  Adjust  the  Polar  Axis.— First  level  the  instru- 
ment carefully  by  the  long  level  of  the  telescope,  using 
in  the  operation  the  tangent  movement  of  the  telescope 
axis  in  connection  with  the  leveling  screws  of  the  parallel 
plates  until  the  bubble  will  remain  in  the  centre  during 
a  complete  revolution  of  the  instrument  upon  its  axis. 

Place  the  equatorial  sights  on  the  top  of  the  blocks  as 
closely  as  is  practicable  with  the  distinct  view  of  a  distant 
object;  and  having  previously  set  the  decimation  arm  at 
zero,  sight  through  the  interval  between  the  equatorial 
sights  and  the  blocks  at  some  definite  point  or  object,  the 
declination  arm  being  placed  over  either  pair  of  the  cap- 
stan-head screws  on  the  under  side  of  the  disk. 

Keeping  the  declination  arm  upon  the  object  with  one 
hand,  with  the  other  turn  the  instrument  half  around  on 
its  axis,  and  sight  upon  the  same  object  as  before.  If  the 
sight  strikes  either  above  or  below,  move  the  two  cap- 
stan-head screws  immediately  under  the  arm,  loosening 
one  and  tightening  the  other  as  may  be  needed  until  half 
the  error  is  removed. 

Sight  again  and  repeat  the  operation,  if  needed,  until 
the  sight  will  strike  the  same  object  in  both  positions  of 
the  instrument,  when  the  adjustment  of  the  axis  in  one 
direction  will  be  complete. 

Now  turn  the  instrument  at  right  angles,  keeping  the 
sight  still  upon  the  same  object  as  before;  if  it  strikes  the 
same  point  when  sighted  through,  the  axis  will  be  truly 
vertical  in  the  second  position  of  the  instrument. 


THE  SOLAB  ATTACHMENT.  85 

If  not,  bring  the  sight  upon  the  same  point  by  the  other 
pair  of  capstan-head  screws  now  under  the  declination 
arc,  reverse  as  before,  and  continue  the  operation  until 
the  same  object  will  keep  in  the  sight  in  all  positions., 
when  the  polar  axis  will  be  made  precisely  at  right  angles 
to  the  level  and  to  the  line  of  collimation. 

It  should  here  be  noted  that,  as  this  is  by  far  the  most 
delicate  and  important  adjustment  of  the  solar  attach- 
ment, it  should  be  made  with  the  greatest  care,  the  bub- 
ble kept  perfectly  in  the  center  and  frequently  inspected 
in  the  course  of  the  operation. 

(4)  To  Adjust  the  Hour  Arc.— Whenever  the  instru 
ment  is  set  in  the  meridian,  as  will  be  hereafter  described 
the  index  of  the  hour  arc  should  read  apparent  time. 

If  not,  loosen  the  two  flat-head  screws  on  the  top  of  th« 
hour  circle,  and  with  the  hand  turn  the  circle  arounO 
until  it  does,  fasten  the  screws  again,  and  the  adjustment 
will  be  complete. 

To  obtain  mean  time,  of  course  the  correction  of  th< 
equation  for  the  given  day,  as  given  in  the  Nautical  Al- 
manac, must  always  be  applied. 

6.  To  Find  the  Latitude.— First  level  the  instru- 
ment very  carefully,  using,  as  before,  the  level  of  the 
telescope  until  the  bubble  will  remain  in  the  center  dur- 
ing a  complete  revolution  of  the  instrument,  the  tangent 
movement  of  the  telescope  being  used  in  connection  with 
the  leveling  screws  of  the  parallel  plates,  and  the  axis  of 
the  telescope  firmly  clamped. 

Next  clamp  the  vertical  arc,  so  that  its  zero  and  that  of 
its  vernier  coincide  as  near  as  may  be,  and  then  bring 
them  into  exact  line  by  the  tangent  screw  of  the  vernier. 

Then,  having  the  declination  of  the  sun  for  12  o'clock 
of  the  given  day  as  affected  by  the  meridianal  refraction 
carefully  set  off  upon  the  declination  arc,  note  also  the 
equation  of  time,  and  fifteen  or  twenty  minutes  before 
noon,  the  telescope  being  directed  to  the  north,  and  the 


86  A  MANUAL  OF  LAND  SURVEYING. 

object-end  lowered  until,  by  moving  the  instrument  upon 
its  spindle  and  the  decimation  arc  from  side  to  side,  the 
sun's  image  is  brought  nearly  into  position  between  the 
equatorial  lines.  Now  bring  the  declination  arc  directly 
in  line  with  the  telescope,  clamp  the  axis  firmly,  and  with 
the  tangent  screw  bring  the  image  precisely  between  the 
lines  and  keep  it  there  with  the  tangent  screw,  raising  it 
as  long  as  it  runs  below  the  lower  equatorial  line,  or  in 
other  words,  as  long  as  the  sun  continues  to  rise  in  the 
heavens. 

When  the  sun  reaches  the  meridian,  the  image  will  re- 
main stationary  for  an  instant  and  then  begin  to  rise  on 
the  plate. 

The  moment  the  image  ceases  to  run  below  is  of  course 
apparent  noon,  when  the  index  of  the  hour  arc  should 
indicate  XII,  and  the  latitude  be  determined  by  the  read- 
ing of  the  vertical  arc. 

It  must  be  remembered,  however,  that  the  angle 
through  which  the  polar  axis  has  moved  in  the  operation 
just  described  is  measured  from  the  zenith  instead  of  the 
horizon  as  in  the  ordinary  solar,  so  that  the  angle  read  on 
the  vertical  limb  is  the  complement  of  the  latitude. 

The  latitude  itself  is  readily  found  by  subtracting  this 
angle  from  90°;  thus,  at  Troy,  the  reading  of  the  limb 
being  found  as  above  directed  to  be  47°  16',  the  latitude 
will  be  90°— 47°  16'  =  42°  44'. 

It  wTill  be  noticed  that  with  this  apparatus  the  latitude 
of  any  place  can  be  most  easily  ascertained  without  any 
index  error,  as  in  the  usual  solar  compass. 

?•  To  Run  Lines  with  the  Solar  Attachment.— 
Having  set  off  the  complement  of  the  latitude  of  the 
place  on  the  vertical  arc,  and  the  declination  for  the; 
given  day  and  hour,  as  in  the  solar,  the  instrument  being 
also  carefully  leveled  by  the  telescope  bubble,  set  the 
horizontal  limb  at  zero  and  clamp  the  plates  together, 
loosen  the  lower  clamp  so  that 'the  transit  moves  easily 


THE  SOLAR  ATTACHMENT.  87 

upon  its  lower  socket,  set  the  instrument  approximately 
north  and  south,  the  object  end  of  the  telescope  pointing 
to  the  north,  turn  the  proper  solar  lens  to  the  sun,  and 
with  one  hand  on  the  plates  and  the  other  on  the  revolv- 
ing arm,  move  them  from  side  to  side  until  the  sun's 
image  is  brought  between  the  equatorial  lines  on  the  sil- 
ver plate. 

The  lower  clamp  of  the  instrument  should  now  be  fast- 
ened and  any  further  lateral  movement  be  made  by  the 
tangent  screw  of  the  tripod.  The  necessary  allowance 
being  made  for  refraction,  the  telescope  will  be  in  the 
true  meridian,  and  being  undamped,  may  be  used  like  the 
sights  of  the  ordinary  solar  compass,  but  with  far  greater 
accuracy  and  satisfaction  in  establishing  meridian  lines. 
Of  course  when  the  upper  or  vernier  plate  is  undamped 
from  the  limb,  any  angle  read  by  the  verniers  is  an  angle 
from  the  meridian,  and  thus  parallels  of  latitude  or  any 
other  angles  from  the  true  meridian  may  be  established 
as  with  the  solar  compass. 

-  The  bearing  of  the  needle,  when  the  telescope  is  on  the 
meridian,  will  also  give  the  declination  of  the  needle  at 
the  point  of  observation. 

The  declination  of  the  needle  being  set  off,  the  needle 
kept  then  at  zero,  or  *'  with  the  sun,"  lines  may  be  run 
by  the  needle  alone,  when  the  sun  is  obscured. 

The  sun,  however,  must  ever  be  regarded  as  the  most 
reliable  guide,  and  should,  if  possible,  be  taken  at  every 
station. 


88  A  MANUAL  OF  LAND  SURVEYING. 

CHAPTEE  IV. 
MEASUREMENT  OF  ANGLES. 

1.  The  instruments  already  described  are  used  both 
for  running  lines  and  for  measuring  angles.    The  transit 
is  used  where  the  greatest  degree  of  accuracy  is  required 
and  where  angles  are  to  be  measured  within  1'  or  less. 

The  compass  is  used  when  no  great  degree  of  accuracy 
is  required  and  the  measurement  of  an  angle  within  5' 
is  as  close  as  is  ordinarily  expected. 

Professional  Surveyors  are  provided  with  the  compass 
or  transit  in  some  of  their  various  forms. 

Students  and  others  may  or  may  not  have  them.  In 
case  of  necessity  the  tape  may  be  used  to  measure  angles, 
and  in  connection  with  the  picket,  sections  of  the  United 
States  Survey  may  be  subdivided,  irregular  fields  meas- 
ured, and  other  similar  operations  performed,  with  a  ra- 
pidity and  accuracy  equal  to,  if  not  superior  to  work  done 
with  a  compass,  the  picket  being  used  to  run  the  lines 
and  the  tape  to  measure  both  distances  and  angles. 

2.  To  Measure  Angles  with  the  Tape. 

This  is  most  conveniently  done  with  the  aid  of  tables 
of  trigonometrical  functions  with  which  the  student  is 
supposed  to  be  familiar. 

Prob.  1.  To  lay  off  a  right  angle  from  a  point  p  in  a 
given  line  AB. 


7TL 

FIG.  13. 


MEASUREMENT  OF   ANGLES:  89 

When  the  sides  of  a  triangle  are  to  each  other  as  3, 4 
and  5,  the  angle  between  the  smaller  sides  is  a  right 
angle.  Hence  to  lay  off  a  right  angle  with  the  tape  or 
chain,  stick  a  marking  pin  at  p  and  then  measure  along 
the  line  p  m  =  3  and  stick  another  pin  at  m.  Then  from 
p  as  a  center  with  a  radius  4  and  from  m  as  a  center  with 
radius  5  strike  arcs  intersecting  at  n.  Then  will  mpn 
be  the  required  angle.  If  the  line  pn  is  to  be  prolonged  as 
a  picket  line,  it  will  be  better  to  range  from,  if  longer 
sides,  as  60, 80  and  100  are  used. 

This  is  the  most  useful  of  the  many  methods  of  laying 
off  a  right  angle  with  the  tape,  and  can  be  applied  where 
any  method  can  be.  The  other  methods  are,  for  the  most 
part,  more  curious  than  useful.  The  following  is  one  of 
the  best  of  them: 

2d  Method.  Measure  along  the  line  in  opposite  direc- 
tions from_p  and  stick  pins  in  the  line  at  m  and  w'  mak- 
ing pm  =  pm'.  Then  from  m  and  m'  as  centres  with 
any  radius  greater  than  pm  strike  two  arcs  Intersecting 
at  n.  Mpn  is  the  required  angle. 


FIG.  14. 

Prob.  2.  From  a  point  p  in  a  given  line  'AB  to  run 
a  line  making  any  required  angle  with  the  line  AB. 

1st  Method.  From  p  measure  p  m  equal  to  the  cosine 
of  the  required  angle  and  stick  a  pin  in  the  line  at  m. 
Then  from  m  as  a  centre  with  a  radius  equal  to  the  sine 
of  the  required  angle  and  from  p  as  a  centre  and  radius  r 
strike  arcs  intersecting  at  n.  Then  mpn  will  be  the 
required  angle  and^>  and  n  will  be  points  in  the  required 
line.  If  r  —  100  then  the  lengths  of  cosine  and  sine  are 
used  just  as  taken  from  the  table  of  natural  sines,  only 


90 


A  MANUAL   OF  LAND   SURVEYING. 


changing  the  decimal  point.  Otherwise  the  tabular 
numbers  must  first  be  multiplied  by  the  length  adopted 
for  r. 


FIG.  is. 

2d  Method.  In  a  similar  manner  we  may  use  the 
natural  tangents  and  secants.  From  p  and  m  as  centres, 
with  the  secant  and  tangent  of  the  required  angle  as 
radii,  strike  arcs  intersecting  at  n.  Secants  not  given  in 
the  table  may  be  found  from  the  table  of  natural  sines 

1 
by  the  formula  secant  — 


cosine. 


\ 


FIG.  16. 

Example  1.    Lay  off,  by  the  use  of  sines  and  cosines,- 
an  angle  of  36°  28X. 

Solution.—  Let  r  =  100  =  pn.      Then  mn  =  59.44, 
pm  =  80.4. 

Ex.  2.    Lay  off  by  the  use  of  tangent  and  secant,  an 
angle  of  25°  20'. 

Solution.— Let  r  =  100  =  pm.      Then  mn  —  47.34; 
pn  =  110.64. 

Ex  4.   Lay  off  by  each  method,  angles  of  48°  20',  63°  15', 
26°  32',  8°  40', 18°  23',  37°  06',  82°  45'. 


MEASUREMENT   OF  AKGL^a.  91 

3d  Method.  By  chords.  From  the  point  p  as  a, 
centre,  with  any  radius, —  preferably  100,  strike  an  arc 
mx.  Find  the  natural  sine  of  half  the  angle.  Double 
it  for  the  chord.  With  this  distance  as  radius,  from  m 
as  a  centre,  strike  an  arc  intersecting  the  arc  mx  at  n. 
Then  p  and  n  are  points  in  the  required  line  and  mpn 
the  required  angle. 


Example  1.  Having  run  the  line  from  the  east  quarter 
post  of  section  26  north  to  the  section  corner  and  marked 
it  with  a  sufficient  number  of  pickets,  it  is  required  to 
locate  the  centre  line  of  a  highway  commencing  at  the 
quarter  post  and  running  north  22^°  west. 

Solution.— Measure  north  in  the  line  from  the  quarter 
post  the  full  length  of  the  tape  =  100,  stick  a  marking  pin 
m  carefully  in  line,  and  strike  an  arc  to  the  left  around  the 
quarter  post  as  a  centre.  Find  the  sine  of  half  the  angle 
and  double  it.  SL.e  11°  15'  X  2  =  .19509  X  2  =  .39018 
or  correcting  the  decimal  point  39.018.  With  this  dis- 
tance as  a  radius,  from  m  as  a  centre,  locate  the  inter- 
secting point  n  which  is  a  point  in  the  required  line. 

The  student  should  now  select  a  level  plat  of  ground, 
mark  out  a  line  upon  it  with  pickets  and  solve  the  pre- 
ceding examples  or  similar  ones,  on  the  ground,  each  one 
by  the  several  different  methods  and  compare  results, 

Also  set  pickets  at  the  angles  of  a  field  of  three  or  more 
sides  and  measure  the  sides  and  angles  of  the  field. 

3.    To  Measure  Angles  with  the  Compass. 

Set  the  compass  up  at  the  intersection  of  the  lines,  be- 
tween which  the  angle  is  to  be  measured.  Put  the  sights 
in  range  with  one  f  the  lines  and  note  the  reading  of  the 


92  A  MANUAL  OF  lAJtfD  SUEVEYING. 


needle.  Then  put  them  in.  range  with  the  other  line 
and  again  note  the  reading  of  the  needle.  Read 
off  from  the  limb,  or  calculate  the  number  of  de 
grees  passed  over  by  the  needle  between  the  two 
readings.  In  land  surveying,  a  line  traced  out  upon  tlu 
ground  is  termed  a  course  and  the  angle  which  the  link 
makes  with  a  north  and  south  line  is  called  its  bearing  or 
course.  In  compass  work  the  bearings  only  are  taker 
The  angles  between  the  lines  of  the  survey  may  be  com- 
puted therefrom  if  necessary.  They  are  seldom  required- 
In  reading  and  writing  down  the  bearings  it  is  customary 
to  state  first  the  direction  of  the  line  from  which  the 
bearing  is  taken  and  then  the  angl3,  to  the  east  01  west, 
which  the  course  makes  with  that  line,  e.  g.,  North  60 
degrees  West.  South  5  degrees  East.  Written  Nc  60°  W; 
S.  5°  E. 

It  is  customary  in  Land  Surveying  to  refer  all  lines 
to  a  meridian  real  or  assumed.  The  cosine  of  a  bearing 
multiplied  by  the  length  of  its  course  is  called  the 
Latitude. 

The  sine  of  the  bearing  multiplied  by  the  length  of  the 
course  is  called  the  Departure. 

When  desirable  to  find  the  angles  between  two  lines 
from  their  bearings,  they  may  be  computed  as  follows* 

Calling  N.  and  S.  meridianal  lettsrs,  we  have  for  the 
angle  between  two  lines  from  the  same  station,  the  fol- 
lowing: 

PRINCIPLES.  —  1.  When  the  meridianal  letters  are  alike 
and  the  others  unlike,  the  angle  is  the  sum  of  the  bearings. 

(2)  When  the  meridianal  letters  are  unlike  and  the 
others  alike,  the  angle  is  the  supplement  of  the  sum  of  the 
bearings. 

(3)  When  loth  the  meridianal  and  the  other  letters  are 
alike,  the  angle  is  the  difference  of  the  bearings. 

(4)  When  both  the  meridianal  and  the  other  letters  are 
unlike,  the  angle  is  the  supplement  of  the  difference  of  the 
t-earings. 


MEASUREMENT  OF   ANGLES.  93 

Observe  that  the  bearings  are  given  in  their  proper  rel- 
ative direction  with  each  other  and  none  of  them  are 
reversed,  as  S.  E.  when  it  should  be  N.  W. 

Examples.  1.  The  bearings,  of  two  lines  are  N.  60°  W. 
and  N.  3°  E.  What  is  the  angle  between  them? 

Ans.  63°. 

a.  Required  the  angles  between  lines  having  the  fol- 
lowing bearings:  N.  37°  E.  and  S.  26°  E.;  N.  87°  E.  and 
S.  86°  W.;  S.  15°  E.  and  S.  26°  E.  Ans.  117°;  179°;  11°. 

3.  Stake  out  a  triangle  in  the  field  and  take  the  bear- 
ings of  the  sides. 

Find  the  angles  of  the  triangle  and  compare  the  sum 
with  180. 

4.  Stake  out  fields  having  4,  5  and  6  sides.    Take  the 
bearings  and  find  the  angles  between  the  sides. 

4.    To  Correct  Courses  of  Random  Lines. 
CASE  1ST:—  Where  the  line  has  but  one  course. 

Random  lines  as  they  are  usually  called  are  simply 
trial  lines  run  to  find  the  true  line  between  two  fixed 
points  which  are  not  visible  from  each  other.  These 
lines  are  usually  started  from  one  of  the  points  and  run 
as  nearly  in  the  true  direction  as  can  be  estimated.  If 
the  estimate  proves  correct,  and  the  line  strikes  the  point 
aiired  for,  the  random  becomes  the  true  line.  If  not,  the 
perpendicular  distance  from  the  line  to  the  point  is 
measured,  from  which  the  correction  for  the  course  may 
be  computed. 


FIG.  is. 

If  PC  is  made  perpendicular  to  AB  as  is  generally 
the  case  where  randoms  are  run  between  corners  of  the 

CP 

Tnited  States  survey  then  Tan.  CAP  = whence 

AP 


04  A  MANUAL  OF   LAND  SURVEYING. 

the  angle  CAP  is  found,  which  is  the  correction  to  be 
applied  to  the  bearing. 

The  angle  CAP,  when  it  is  quite  small,  may  be  found 
by  multiplying  57.3°  by  PC,  and  dividing  by  AC.  This 
is  called  the  Fifty-seven  and  three-tenths  rule. 
The  rule  depends  upon  the  fact  that  for  small  angles,  AP 
differs  insensibly  from  AC,  and  CP  from  the  arc  sub- 
tending the  angle  CAP. 

Whence,  angle  C^P:360°::CP:2X3.1416X^P, 

CP      360°         OPX57.30         CTX&7.30 

or  angle  CAP  —  — X = ,  or 

AP    6.2832  AP  AC 

The  semi-circumference  of  a  circle,  with  radius  AP,  is 
3.14159265XAP. 

Whence  arc  I'  =  3.14159265  X  AP  -5-  10800. 
If  AP  =  1  ch.,  arc  1'  =  0.00029088  ch.  =  0.029088  1. 

If  AP  =  1  mi.  =  80  ch.,  arc  V  =  0.029088  1.  X  80 
=  2.327  1.  -  2^  1. 

When  angle  PAG  =  V  and  AP  or  AC  =  1  mi.,  the 
perpendicular  PC,  without  perceptible  error,  is  2%  links. 
The  line  PC  is  called  the  departure  of  AC,  for  the  dis- 
tance AP  or  AC. 

Taking  2%  1,  as  the  departure  of  80  ch.  at  an  angle  of 
1',  the  departure  for  40  ch.,  would  be  %  of  2%  1.  =  1£  1. 
=  1  1.  +  i  of  1 1. 

For  quite  small  angles,  the  departure  varies  directly  as 
the  angle.    Whence,  for  40  ch.,  the  following: 
Dep.  for  1'  =  1 1.  +  |  of  1 1. 
"  2/  =  2  1.  +  |  of  2  1. 
«        «  3'  =  3  1.  +  I  of  31. 
and  so  on,  practically  true,  to  60'  or  1°. 

For  any  other  distance,  at  the  same  angle,  the  depar- 
ture varies  directly  as  the  distance.  Accordingly, 

Given  minutes  of  angle,  to  find  links  of  departufe, 
we  have  the  following: 

KULE. — To  the  number  of  minutes,  add  its  one-sixth 
and  multiply  the  sum  by  the  ratio  of  the  distance  to 
40  ch.  (Good  to  sixty  minutes.) 


MEASUREMENT  OF  ANGLES.  95 

On  the  following: 

GENERAL  RULE. — Multiply  0.0291  by  the  number  of 
minutes,  and  multiply  the  product  by  the  number  of 
chains  in  the  distance.  (Good  to  240  minutes.) 

Example. — Given  angle  —  307  and  distance  ±=  23.20  ch., 
to  find  the  departure. 

Since  for  40  ch.,  V  of  angle  gives  \\  1.  of  depar- 
ture, we  may  say,  without  sensible  error  for  a  small  angle 
that  1 1.  of  departure  gives  f  of  lx  of  angle,  for  the  same 
distance. 

Or  as  it  may  be  written, 

Dep.  of  1  i.  =  V  —  \  of  1'. 
Similarly,          "      "  2  1.  =  2'  —  }  of  2', 
"  3 1.  =  3'  —  1  of  3', . 
and  so  on,  practically  true  to  607  or  1°. 

Eor  any  other  distance  with  the  same  departure,  the 

angle  varies  inversely  as  the  distance.    Accordingly, 

/ 

Given  links  of  departure,  to  find  minutes  of  angle, 
we  have  the  following: 

RULE. — From  the  number  of  links  of  departure,  sub- 
tract its  one-seventh  and  divide  the  remainder  by  the  ratio 
of  the  distance  to  40  ch.  (Good  to  60  minutes.) 

GEXERAL  RULE. — Multiply  0.0291  by  the  number  of 
chains  in  the  distance,  and  divide  the  number  of  links  of 
departure  by  the  product.  (Good  to  240  minutes). 

In  the  Table  of  Departures,  the  value  of  PC  in  chains 
and  decimals  is  given  for  angles  from  V  to  60',  and  for 
the  distances  most  commonly  required  in  making  resur- 
veys  and  subdivisions  of  Sections  of  the  United  States 
Survey.  To  use  the  Table:  Having  measured  the  outing 
PC  on  the  ground,  find  the  nearest  tabular  nuwber  in  the 
column  for  the  corresponding  distance. 

The  angle  will  be  found  in  the  minute  column. 

Example  1.  Commencing  at  the  west  quarter  post  of 
Section  16,  and  running  north,  the  random  line  intersected 


96  A   MANUAL    OF   LAND   SURVEYING. 

the  north  line  of  the  section,  15  links  east  of  the  corner 
What  is  the  amount  of  the  correction  for  course  ? 

Solution.  In  40  chain  column,  nearest  number  .151, 
Corresponding  number  of  minutes  13. 

2.  Commencing  at  the  south  quarter  post  of  section  16 
with  declination  of  needle  estimated  at  2°  17'  E.  set  off 
on  the  vernier,  ran  north  on  random  and  intersected  the 
north  line  of  the  section,  42  links  east  of  the  quarter  post. 
What  is  the  declination  of  the  needle  as  referred  to  the 
quarter  line? 

Solution.  Distance  80  chains,  correction  18X.  As  the 
line  came  out  east  of  the  corner,  it  is  evident  that  the 
angle  between  the  magnetic  meridian  and  the  quarter 
line  was  18'  greater  than  was  estimated,  =  2°  35'. 

NOTE.— The  North  and  South  lines  of  the  United  States  Survey 
are,  in  a  legal  sense  all  true  meridians,  whatever  they  may  be  astron- 
omically, and  their  locations  are  fixed  by  the  monuments  planted  for 
the  section  corners  and  quarter  posts.  Hence  it  is  a  custom  amon^ 
Surveyors  to  refer  the  declination  of  the  needle— or  the  variation  as  it 
is  more  frequently  called,  to  these  lines,  and  to  mark  on  each  line 
on  their  plats,  the  declination  for  that  line.  Under  that  custom  the 
line  referred  to  in  Example  2  would  be  marked  Var.  2°  35'  E. 

3.  "East   on   random    between    Sections    13  and   24. 
79.98  chains  intersected  east  boundary  34  links  south  of 
post."    What  is  the  bearing  of  the  corrected  line  running 
west  ?  Am.  S.  89°  45'  W. 

CASE  2ND.—  Where  the  line  is  a  broken  one  of  several 
courses. 

Surveyors  are  frequently  called  on  to  retrace  the  lines  of 
angling  roads  to  settle  the  boundaries  of  adjacent  lands, 
or  to  locate  meander  lines,  or  to  find  the  boundaries  of 
irregular  tracts,  where  several  courses  have  to  be  run 
between  the  nearest  known  points  of  the  original  survey. 

In  such  cases  random  lines  are  run  according  to  the 
notes  of  the  original  survey,  and  temporary  stakes  driven 
at  the  angles  of  the  random  line.  It  will  generally  be 
found  that  corrections  for  course  or  distance  or  for  both 
will  have  to  be  made  to  place  the  stakes  in  their  correct 
location. 


MEASUREMENT  OF  ANGLES.  97 

PROBLEM. — To  correct  a  random  Hue  of  several  courses. 

In  Fig.  19  let  A,  B,  C,  D  represent  the  lines  and  angles 
of  the  original  survey  between  the  known  points  A  and  D. 


FIG.  19. 

Let  Df  represent  the  terminus  of  a  random  run  to  re- 
trace these  lines,  the  direction  and  distance  of  which 
from  D  is  known. 

From  A  draw  the  line  AD,  producing  it  indefinitely 
beyond  D ;  also,  from  A  as  a  centre,  with  radius  AD,  draw 
an  arc  through  D.  ^N"ow,  if  tlie  error  in  the  random  was 
of  direction  only,  then  the  point  D'  would  be  in  the  arc. 
If  it  was  an  error  of  the  chain  only,  Df  would  be  in  the 
line  AD  or  AD  produced.  Hence  the  position  of  D/  with 
reference  to  the  arc  and  the  line  AD  indicates  the  kind 
of  correction  and  in  what  direction  it  is  to  be  applied. 
AD 

is  the  length  of  the  original  chain  in  terms  of  the 

ADf 

chain  used  on  the  random.  That  portion  of  the  arc 
which  is  intercepted  between  the  point  D  and  a  line 
joining  AD',  measures  the  angle  of  correction.  In  the 
field  we  may  calculate  the  course  and  length,  and 
run  a  sufficient  part  of  the  line  D'A,  and  then  trace  the 
arc  from  D  to  its  intersection  with  that  line,  and  thus 
find  the  relative  length  of  the  lines  AD  and  AD't  by  which 
to  determine  the  correction  for  the  chain  and  also  find 
the  chord  of  tke  angular  correction;  or  they  may  be 
calculated  as  shown  in  the  following  example: 

Example  1.  —  The  boundaries  of  a  farm  between  the 
nearest  known  monuments  are  as  follows,  (See  Fig.  19): 

1.  N.  16°,  E.  12.00  chains. 

2.  N.  72°,  E.  26.00       " 

3.  S.  22°,  E.  14.00       " 


98 


A  MANUAL   OF   LAND   SURVEYING. 


A  random  was  run  with  var.  2°  30'  E.  and  came  out 
N.  28°  E.  32  links  from  the  monument.  Required  the 
correction  for  the  variation  of  needle  and  for  the  stakes 
in  the  angles  of  the  random  line. 

We  will  first  find  the  total  latitudes  and  departures  of 
each  station  on  the  random  line,  and  the  direction  and 
distance  of  a  line,  AD',  which  will  join  the  termini. 


N.  Lat. 

S.  Lat. 

E.  Bep. 

Tot.  Lat. 

Tot.  Dep. 

1.    N.  16°  E.  12.00 
2.    N.  72    E.  26.00 
3.     S.  22    E.  14.00 

11.54 
8.03 

12.98 

3.31 
24.73 
5.24 

11.54 
19.57 
6.59 

3.31 

28.04 
33.28 

If  we  now  divide  the  total  departure  of  the  point  I)' 
by  its  total  latitude  we  will  have  the  tangent  of  the 
bearing  of  the  line  D'A. 
33.28 

. =  5.050  =  tan  78°  48'  or  S.  78°  48'  W. 

6.59  

The  length  of  the  line  D'A  =  j/6.592  +  33. 282  ==  33.927, 

If  we  now  subtract  the  bearing  of  the  line  D'D  from 
the  bearing  of  the  line  D'A  we  shall  have  the  angle 
DD'A  =  78°  48'  —  28°  =  50°  48'.  Let  DH  be  a  perpen 
dicular  from  D  to  the  line  AD' ;  then  we  have  the  follow 
ing  equations: 

D H = D'D  sin  AD'D  =  .  32  X  •  77494  =  .  24798-f-. 
D'H  =  D'D  cos  AW  =  .  32  X '  •  63203  =  .  20225. 
AH  =  A  D'  —  D'H  =  33 . 926  —  .  20225  =  33 . 7237+. 
DH 

=  tan  DAD'  =  .24798+  •+•  33.7237+  --=  .00735  =- 

AH 
tan  25'  =  correction  for  course. 

. AH 

AD  =  y AH'2  +  HD2  =  —        —  =  33. 7237 -r-  .99997  = 

cos  DAD' 

33. 724.    When  the  angle  DAD'  is  small,  AD  and  AH  may 
be  considered  equal,  without  sensible  error. 
AD       33.724 

= =  .99404  =  length  of  original  chain  in 

AD'      33.926 

terms  of  the  chain  used  on  the  random.    As  the  randon 


MEASUREMENT  OF  ANGLES. 


99 


came  out  to  the  left  of  the  true  line  the  variation,  2°  3(X  E., 
was  too  great,  hence  we  subtract  the  25',  giving  2°  05' 
as  the  variation  of  the  needle  from  the  meridian  of  the 
original  survey.  To  find  corrections  for  the  stakes  it  will 
be  better  to  refer  them  to  the  meridian  of  the  random, 
hence  we  will  now  apply  the  corrections  for  course  and 
distance  to  find  the  courses  and  distances  of  the  original 
survey,  as  they  would  be  according  to  the  meridian  and 
measure  of  the  random.  This  done,  we  calculate  their 
total  latitudes  and  departures.  The  difference  between 
these  arid  the  latitudes  and  departures  of  the  correspond- 
ing points  of  the  random  is  the  correction  to  be  applied. 


N.  Lat. 

S.  Lat. 

E.  Dep. 

Tot.  Lat. 

Tot.  Dep. 

1.  N.  16°25'E.  11.928 

11.44 

3.37 

11.44 

3.37 

2.  N.  72  25  E.  25.844 

7.81 

24.64 

19.25 

28,01 

3.  S.  21  35  E.  13.916 

12.94 

6.12 

6.31 

33.12 

The  last  course  is  computed  in  this  table  simply  as  a 
check  on  the  work,  as  it  was  a  condition  of  the  problem 
that  the  line  LDf  was  N.  28°,  E.  32  links;  from  which  it  is 
known  that  the  difference  between  the  two  points  is: 
latitude  28  Iks.,  and  departure  15  Iks.  We  will  now  com- 
pare the  results  in  the  two  tables  and  find  the  correction 
at  S,  C  and  D. 


B 

Lat.       Dep. 

c 

Lat.         Dep. 

D 
Lat.         Dep. 

Random  Line  
Original  Line- 

11.54         3.31 
11.44         3.37 

19.57         28.04 
19.25         28.01 

6.59         33.28 
6.31         33.13 

Correction  

S.  10        E.  6  |  S.  32         W.  3 

S.  28        W.  15 

Example  2.  —  Description  of  a  highway  between  two 
known  points: 

1.  N.62°  E.  14.  00  chains. 

2.  K43^°  E.8.00     " 

3.  K  5 


°  W.  12.00  " 
K72^°  E.  10.25  " 
S.  12°  W.  6.43  " 


A  random  run  with  var.  2°  17'  E.  came  out  62  Iks.  east 
of  the  point.     What  is  the  correction  for  variation  of 


100 


A   MANUAL   OF   LAND   SURVEYING. 


needle,  and  what  change  must  be  made  in  the  position  of 
each  stake  at  the  angles  of  the  random  ? 

5.    To  Measure  Angles  with  the  Transit. 

1.  Set  up  the  transit  at  the  apex  of  the  angle  and  set 
the  zero  of  the  vernier  to  coincide  with  the  zero  of  the 
limb.    Clamp  the  plates  in  this  adjustment  and  with  the 
clamp  to  the  spindle  loosened,  turn  the  telescope  in  the 
direction  of  one  of  the  lines.    Clamp  the  spindle  and 
bring  the  wire  exactly  to  centre  the  line  by  the  slow 
motion  screw  to  the  spindle  clamp.    Unclamp  the  vernier 
and  turn  the  telescope  in  the  direction  of  the  other  line. 
Clamp  the  vernier  in  that  position  and  make  the  final 
adjustment  of  the  wire  to  the  line  by  the  use  of  the  upper 
tangent  screw.    The  angle  may  then  be  read  from  the 
limb. 

2.  Instead  of  first  setting  the  verniers  at  zero  they  may 
be  clamped  in  any  position  on  the  limb  and  then  the  differ- 
ence in  the  two  readings  will  be  the  angle.    When  great 
accuracy  is  required  numerous  readings  of  the  angle  me 
taken  on  various  parts  of  the  limb  and  the  mean  of  the 
several  results  taken  for  the  final  reading. 

3.  To  find  the  angle  which  the  parts  of  a  broken  line 
form  with  any  given  line. 


i: 


FIG.  20. 


MEASUREMENT  OF  ANGLES. 


101 


SUGGESTIONS.— Let  ABCDEF  be  a  broken  line,  and 
suppose  it  is  required  to  find  the  angles  which  the  parts 
BC,  CD,  DE  and  EF  form  with  the  line  AB. 

Set  the  transit  at  B,  with  the  vernier  set  at  zero. 
Loosen  below,  reverse  the  telescope  and  direct  H  to  4. 
Clamp  the  limb,  revolve  the  telescope  in  its  horizonta"- 
axis,  unclamp  the  vernier  and  direct  the  telescope  to  C. 
The  reading  of  the  instrument  will  be  the  angle  bBC 
the  line  which  BC  forms  with  the  line  AB. 

Remove  to  C;  and,  leaving  the  vernier  clamped,  un- 
clamp below,  reverse  the  telescope,  and  direct  it  to. 2?. 

The  limb  remaining  securely  clamped,  revolve  the 
telescope,  unclamp  the  vernier,  and  direct  to  D.  The 
reading  will  now  be  the  angle  cCD  which  the  line  CD 
forms  with  the  line  Co  or  its  parallel  AB. 

The  work  goes  on  in  this  manner  to  its  close. 
Let  the  student  further  describe  it. 

If  the  broken  line  enclose  a  field,  the  reading  of  the 
instrument  when  set  as  at  A  and  directed  to  B,  having 
gone  entirely  around  the  field,  should  be  360°.  This  con- 
stitutes a  check  against  errors  occu_Ting  anywhere  in  the 
work. 

4.    To  measure  an  angle  of  elevation  or  depression. 

c 


A 


FIG.  21. 

SUGGESTIONS.  — Set  the  instrument  at  the  vertex  of 
the  angle  and  level  the  horizontal  limb. 


102  A   MANUAL    OF   LAND    SURVEYING. 

Revolve  the  telescope  upward  or  downward  as  the 
case  may  require,  and  adjust  the  line  of  sight  to 
the  inclined  sivie  of*  the  angle.  Take  the  reading  of  the 
vertical  circle,  applying  the  proper  correction  for  index 
errcr- 

Otherwise,  take  the  reading  of  the  circle,  repeat  the 
observation  with  the  telescope  and  vernier  plate  reversed, 
and  find  the  mean  of  the  two  readings  for  the  angle 
sought. 

6.  Verniers  are  auxiliary  scales  for  measuring 
smaller  portions  of  space  than  those  into  which  the 
main  scale  is  divided.  They  are  movable  beside  the 
main  scale  and  are  divided  into  parts  which  are  either 
a  little  shorter  or  a  little  longer  than  the  parts  into  which 
the  main  scale  is  divided.  This  small  difference  in 
length  is  what  we  are  enabled  to  measure. 

When  the  limb  of  a  transit  is  divided  to  half  degrees  it 
is  common  to  make  either  29  or  31  divisions  of  the 
Vernier  Scale  equal  to  30  on  the  limb,  making  each 
division  on  the  vernier  31'  or  29'  in  length. 

The  zero  of  the  Vernier  Scale  is  the  point  to  which  the 
reading  is  to  be  taken.  Suppose  the  zero  line  of  the  vernier 
to  make  a  straight  line  with  some  even  division  of  the  limb 
and  each  division  on  the  vernier  scale  is  297  in  length. 
Xow  if  the  Vernier  be  moved  ]x,  the  first  line  of  the 
Vernier  Scale  from  zero  in  the  direction  in  which 
the  vernier  was  moved,  will  be  in  a  line  with  the  first 
division  on  the  limb.  If  moved  2'  the  second  lines 
will  coincide;  if  3X  the  third  lines  ;  and  so  on  to  the 
end  of  the  scale.  Such  a  vernier  is  called  direct  reading. 
It  is  the  kind  most  commonly  used  on  surveyors'  instru- 
ments. 

Suppose  however  that  the  spaces  on  the  vernier  were 
31'  long.  Then  when  the  vernier  was  moved  forward  V 
the  first  line  back  of  the  zero  point  would  coincide  with 
the  line  in  the  limb  and  so  on.  Such  a  vernier  is  called  a 
retrograde  vernier. 


MEASUREMENT  OP   ANGLES.  103 

To  read  any  vernier.  If  the  zero  of  the  vernier  coin 
cides  with  any  division  of  the  scale,  that  will  be  the  cor- 
rect reading.  If  not,  note  the  nearest  next  less  division 
on  the  limb,  and  then  look  along  the  vernier  scale  till  a 
line  is  found  which  coincides  with  a  line  on  the  limb. 
The  number  of  this  line  on  the  vernier  tells  that  so  many 
of  the  subdivisions  which  the  vernier  indicates  (usually 
minutes)  are  to  be  added  to  the  reading  of  the  entire 
divisions  on  the  limb. 

If  several  lines  appear  to  coincide  equally  well,  take  the 
middle  line. 


A04 


A  MANUAL  OF  LAND  SURVEYING. 


CHAPTER  V. 

PASSING  OBSTACLES.    MEASURING  INACCESSIBLE 
DISTANCES. 

Having  considered  the  various  methods  of  running 
lines  and  measuring  angles  we  are  now  prepared  to  take 
up  some  further  problems  in  passing  obstacles  in  the 
line  and  measuring  inaccessible  distances. 

These  problems  may  be  solved  in  the  field  by  the  use  of 
the  picket  and  tape,  the  compass,  or  the  transit. 

1.  To  pass  an  obstacle  in  the  line  and  measure,  the 
distance. 

1st,  by  Parallel  Lines.  Prom  a  in  the  line  AB  run 
and  measure  the  line  ac  in  any  convenient  direction,  a 
sufficient  distance.  From  c  run  cd  parallel  with  AB. 


FIG.  22. 

Prom  d,  run  and  measure  db  equal  to  and  parallel  with 
ac.  Then  ab  =  cd  and  b  is  a  point  in  the  line  AB. 
When  running  through  heavy  forests  or  towns  it  will 
often  be  necessary  to  run  several  parallel  lines  before 
returning  to  the  original  line. 

2.  By  6O°  Angles.  From  a  run  and  measure  ac 
making  the  angle  Bac  =  60°.  Run  and  measure  cb  ^  ac 
and  the  angle  acb  —  60°.  Then  6  is  a  point,  in  the  line 


PASSING  OBSTACLES. 


105 


AS  and  the  angle  abc  —  60°,  whence  the  line  may  be  con- 
tinued; ab  will  equal  at: 


FIG.  23. 
2.    To  Measure  Inaccessible  Distances. 

CASE  IST.    When  the  points  are  visible  from  each  other 
as  ovei'  a  stream  or  pond. 


FIG.  24. 

I.    By  Similar  Triangles. 

From  a  point  a  in  the  line  AS,  required  the  distance 
ab  across  the  stream. 

At  a  erect  a  perpendicular  ac  to  the  line  AB.  From  c 
run  a  perpendicular  to  cb  intersecting  AS  at  d.  Measure 
ac  and  ad.  Then  as  the  triangles  cad  and  bed  are  similar, 

ac2 

ad  :  ac  =  ac  :  ab,  whence  ab  =  — . 

ad 

There  are  numerous  other  devices  for  obtaining  the 
distance  ab  by  similar  triangles  on  the  ground.  Let  the 
student  work  out  some  of  them  in  the  field. 


106 


A  MANUAL  OF   LAND   SURVEYING. 


2.  Method  by  Tangents. 
Erect  a  perpendic- 
ular to  AB  at  a  and 

run  it  a  sufficient 
distance  ac.  Meas- 
ure the  angle  acbt 
Then  ab  =  ac  X  tan 
acb.  If  «*c  is  made 

100  or  1,000,  a&  may  be  read  directly  from  the  table  of 
natural  tangents,  observing  to  put  the  decimal  point  in 
the  proper  place.  If  acb  =  45°  then  ab  =  ac. 

3.  Method  by  Sines. 

From  a  run  a  line 
ac  as  most  conven- 
ient. Measure  the 
angles  acb  and  cab 
and  the  side  ac.  Com- 
pute the  angle  abc. 
Then  sin  abc  :  sin  acb 


ab 


ab  = 


ac  sin  acb 


FIG.  26. 


4.  Method  by  Cosines. 

From  a  run  a  line 
ac  to  the  point  c  in  a 
line  perpendicular  to 
AB  at  b. 

Measure  the  angle 
cab  and  the  line  ac. 

Then  a&  =  ae  X  cos 

5.  Method  by  Secants. 

Run  ac  as  be- 
fore, to  a  point  c 
from  which  a  per- 
pendicular to  ac 
will  strike  the 
the  point  b.  Meas- 
ure ac  and  the  an- 


FIG.  28. 


INACCESSIBLE  DISTANCES. 


107 


gle  bac.    Then  ab  =  ac  X  secant  bac.   Ifac=  100  or  1,000 
the  distance  ab  is  taken  directly  from  the  table. 

6.    By  5°  43'  Angle. 

From  a  lay  off  the 
angle  bac  =  5°  43', 
making  be  perpen- 
dicular to  ab.  Meas- 
ure be.  Then  ab  =  FIG.  29. 
lObc.  This  method  gives  results  too  large  by  1.07  in  1,000. 

CASE  2ND.    Where  the  points  are'  irivisibJe  from  each 
other. 

1.  If  visible  and 
accessible  from  a 
common  point  c 
outside  the  line. 
Measure  the  lines 
ac  and  be  and  the 
angle  acb.  Sub-  FIG.  30. 

tract  this  angle  from  180°  and  we  have  the  sum  of  the 
remaining  angles  of  the  triangle,  to  find  the  difference. 

abc  -f  bac          abc  —  bae 
tan :  tan 


Then  ac  +  be  :  ae  —  6c 


abe~-\-  bac       abe  —  bae 

And  — 1 

2  2 

abc  -f  bac      abc  —  bac 


=  abc. 


Also 


=  bac. 


2  2 

ab  =  acX  cos  bac  +  bcX  cos  abc. 

If  a  and  6  are  inaccessible  from  c,  the  sides  ac  and  be 
may  be  measured  by  any  of  the  preceding  methods. 


2.  If  instead  of 
two  lines  ae  and  bct 
we  have  a  broken 
line  of  any  num. 
ber  of  courses,  as 
abcdef,  thebear- 


FlG.  31. 


108  A   MANUAL  OF   LAND   SURVEYING. 

ings  of  which  are  referred  to  the  line  of  as  a  meridian 
—then  the  algebraic  sum  of  the  products  of  the  cosines 
of  the  several  bearings  into  their  respective  distances 
will  be  equal  to  of. 

In  the  United  States  Surveys  distances  across  lakes 
and  bends  of  large  streams  are  frequently  computed  from 
the  latitudes  and  departures  of  the  courses  around  them. 

Examples.— I.  In  Fig.  24  ac  =  100  ad  =  27.  Required 
ab.  Am.  370.37+ 

2.  Same  Figure,  ac  =  250,  ad  =  96.    Required  ab. 

Ans.  651.04+ 

3.  Fig.  25,  ac  =  100,  angle  c  =  61°  20'.    Required  ab. 

Ans.  182.9. 

4.  Same  Figure,  ac  =  250,  angle  c  =  61°  10'.   Required 
ab.  Ans.  454.1+ 

5.  Fig.  26,  ac  =  500,  angle  a  =  48°  20',  angle  c  =  118° 
1(X.    Required  ab.  Ans.lSSS.l 

6.  Same  Figure,  ac  =  658,  a  —  54°  16',  c  =  88°  32'. 
Required  ab.  Ans.  1087.9+ 

7.  Fig.  27,  ac  =  1,000,  angle  a  =  28°  35'.    Required  ab. 

Ans.  878.12+ 

8.  Same  Figure,  ac  =  950,  angle  a  =  18°  56'.  Required 
ab.  Ans.  898.6. 

9.  Fig.  28,  ac  =  100,  angle  a  =  76°  40'.    Required  ab. 

Ans.  433.6+ 

10.  Same  Figure,  ac  =  250,  angle  a  =  56°  20'.  Required 
ab.  Ans.  450.97. 

11.  Fig.  29,  ac  =  900,  be  =  648,  angle  c=  112°.     Re- 
quired ab.  Ans.  1291. 

12.  Given  the  following  courses  and  distances  along  a 
broken  line  between  the  points  a  and  b.    Required  the 
distance  ab. 

1.  N.  18°  E.  6.25  chains. 

2.  N.  40°  E.  8.00      « 

3.  N.  5°  W.  12.00      " 

4.  K.  44°  W.  8.68     "  Ans.  30.26+  chains. 


INACCESSIBLE   DISTANCES. 


109 


±3.  The  field  notes  of  the  meanders  of  a  lake  in  sec- 
tions 11  and  12  in  the  township  1,  south,  range  10  west, 
meridian  of  Michigan,  —  by  the  government  survey,  read 
as  follows: 


Courses 

X.  58  E. 

N.11W. 
X.63W. 

Chs.  Lks. 

10.00 
20.00 
5.16 

Began  at  post  in  line  of  sections  11  and  12  on 
south  side  of  lake  :  thence  in  sec.  12. 

to  post  in  line  of  sec.  11  and  12,  N.  side  of  lake. 

X.63W. 
S.  GO  W. 
S.  14  E. 
S.  33  W. 

S.  r>i  E. 
N.73V4  E 

5.00 
6.00 
10.00 
15.00 
10.00 
7.90 

in  section  11. 
to  place  of  beginning. 

Kequired  the  distance  between  the  posts  on  the  oppo- 
site sides  of  the  lake.  Compute  the  distance  by  the  mean- 
ders on  each  side  of  the  lake.  Compare  the  results  to- 
gether, and  also  with  the  distance  returned  in  the  field 
notes  which  is  27.27  chains. 

14.  There  is  a  cliff  beside  a  railroad  in  the  Wasatch 
Mountains  known  as  the  Castle  Gate.  Desiring  to  know 
its  height  above  the  railroad  grade  I  set  up  the  transit 
at  Station  744  of  the  railroad  survey  and  took  the  angle 
of  elevation  to  the  top  of  the  cliff  =  38°  42'.  Elevation  of 
station  744  =  6573.62  ft 

Height  of  instrument  above  station  744  =•  4.84  ft. 

I  next  went  to  station  748  in  the  line  with  and  400  ft. 
farther  away  from  the  cliff  and  again  took  the  angle  of 
elevation  to  the  top  of  the  cliff  =  26°  15'. 

Elevation  of  station  748  ==  6567.62  ft. 

Height  of  instrument  above  the  station,  4.56  ft. 

Required  the 
height  of  the  Cas- 
tle Gate  above  the 
station  744  and  its 
horizontal  dis- 
tance. 
Atiswer. 

Height  501.54. 

Distance  620. 


110  A   MANUAL   OF   LAND   SURVEYING. 

15.  On  Christmas  1881  a  party  of  surveyors  climbed  a 
mountain  peak,  erected  a  monument  on  its  summit  and, 
named  it  Christmas  Peak.  Observations  from  the  line 
of  the  railroad  survey  were  made  as  follows,  the  stakes  of 
that  survey  being  100  feet  apart: 

From  station  933  +  49.6  P.  T. 

Angle  of  elevation  of  summit,  23°  42'. 

Angle  to  right  from  railroad  line  ahead,  76°  1CK. 

Elevation  of  station,  5005.28  ft. 

Instrument  above  station,  4.82  ft. 

From  station,  940  +  31.4  P.  C. 

Angle  to  left  from  railroad  line  back  =  82°  18'.  lie- 
quired  the  height  of  the  peak  and  its  distance  from  sta- 
tion 933  +  49.6. 


3.   Other  Methods  of  Measuring  Distances, 

1.  To  Gross  a  stream  or  pond. 

Set  up  the  transit  at  a  convenient  point,  a.    Set  up  a 

rod  at  &  in  the  line, 
at  a  convenient  dis- 
tance, as  100  feet, 
from  a.  Set  up  a 
second  rod  in  line  at 
c,  over  the  stream* 
Any  plain,  straight 

rods  will  answer.  Leveling  rods  with  targets  are  conve- 
nient. They  should  be  set  up  plumb.  Mark  points  d  and 
e,  in  line,  on  the  rods  where  the  horizontal  wire  of  the 
telescope  cuts  them,  liaise  or  lower  the  telescope  and 
mark  two  other  points, /and  g,  in  line  on  the  rods  where 
the  wire  cuts  them.  Measure  df  and  eg.  Then  adf  and 
aeg  are  similar  triangles,  and  df. :  of : :  eg  :  ag.  If  df~  1 
and  af=  100,  eg  =  6.25;  then  ag  =  625. 

2.  Stadia  Measures. 

1.   Instead  of  using  two  rods  as  described  in  the  last 
paragraph,  two  wires  are  sometimes  placed  in  the  dia- 


INACCESSIBLE  DISTANCES.  Ill 

phragm  of  the  telescope  and  adjusted  at  such  a  distance 
apart  that  they  will  cover  a  specified  space  on  a  rod,  as 
1  foot  when  the  rod  is  100,  200  or  any  other  specified  dis- 
tance away.  These  wires  are  one  on  each  side  of  and 
parallel  with  the  horizontal  wire  of  the  telescope.  They 
may  be  either  fixed  on  the  diaphragm  or  attached  to  slides 
by  which  their  distance  apart  may  be  adjusted.  When 
the  wires  are  adjusted  to  cover  a  certain  space,  as  one 
foot  on  a  rod  placed  100  feet  away,  they  will  cover  two 
feet  on  a  rod  200  feet  away,  or  .5  foot  on  a  rod  50  feet 
away.  This  proportion  is  strictly  true  only  when  the 
measures  are  taken  from  a  point  in  front  of  the  in- 
strument at  a  horizontal  distance  from  the  object  glass 
equal  to  its  focal  length.  The  focal  length  may  be  found 
nearly  enough  by  measuring  from  the  plane  of  the  object 
glass  to  the  capstan-headed  screws  which  carry  the  dia- 
phragm. When  the  telescope  is  focused  on  some  very 
distant  object,  as  the  moon  or  a  star,  the  horizontal  dis- 
tance from  the  plumb  line  to  the  point  mentioned  forms 
a  constant  which  is  to  be  added  to  all  the  distances  as 
taken  from  the  rod. 

2.  It  is  more  convenient,  though  less  accurate,  to  adjust 
the  wires  so  that  they  will  cover  the  required  space  on 
the  rod  at  a  specified  distance  measured  from  the  center 
of  the  instrument.  This  method  is  usually  adopted  on 
the  government  surveys,  where  stadia  measures  are  taken, 
the  length  of  the  base  being  taken  at  about  a  mean  of 
the  distances  which  the  stadia  is  intended  to  measure. 
For  all  shorter  distances  the  reading  is  too  small.  For 
longer  distances  it  is  too  large.  The  error  is  neglected 
as  of  no  consequence  in  the  class  of  work  for  which  the 
stadia  is  used. 

When  the  stadia  wires  are  not  adjustable  the  rod  is 
graduated  to  conform  to  the  wires.  A  rod  is  set  up  at 
the  selected  distance  from  the  transit.  The  space  inter- 
cepted on  it  by  the  wires  is  subdivided  decimally,  and 
the  stadia  rod  graduated  to  that  scale. 

Where  the  wires  are  adjusted  to  cover  a  foot  on  a  rod 


112 


A   MANUAL   OF   LAND   SURVEYING. 


100  or  200  feet  away,  the  ordinary  leveling  rod  answers 
the  purpose  of  a  stadia  rod. 

3.  In  case  the  measures  are  not  on  horizontal  lines  it 
will  be  necessary  to  apply  a  correction  to  the  stadia  read- 
ings to  reduce  them  to  the  horizontal.  If  the  rod  has 
been  held  perpendicular  to  the  line  of  sight,  the  horizontal 
distance  is  found  by  multiplying  the  distance  to  the  rod 
by  the  cosine  of  the  angle  of  elevation  or  depression. 

The  position  of  the  rod  is  determined  either  by  a  right- 
angle  sight  applied  to  the  rod,  or  by  the  rodman  slowly 
moving  the  top  of  the  rod  back  and  forth  until  the 
smallest  intercept  is  obtained.  On  hillsides  it  will  be 
found  quite  as  easy  to  hold  the  rod  perpendicular  to  the 
line  of  sight  as  to  hold  it  plumb. 

When  the  rod  is  held  plumb  and  the  base  is  measured 
from  the  point  in  front  of  the  transit  the  reduction  to 
horizontal  is  made  as  follows: 
Let/=  focal  distance  of  the  telescope, 

r  =  space  intercepted  on  the  rod  as  held  vertically, 
s  =  image  of  the  same  intercepted  by  the  stadia 
wires, 

CO'  =  line  of  sight  at  an  angle  e  with  the  horizon. 

»  Let  A'B'  =  r' 
be  the  intercept 
on  the  rod  as  in- 
clined at  an  angle 
e  with  the  vertical; 

rf 
and  let  b'  =f—  be 


the  corresponding 
base.  Let  the  an- 
o  gle  O'CB  or  O'CA 
=  v.  We  shall 
then  have : 


FIG.  34. 


Angle  OCB  =  e  -f-  v,  and  angle  OCA  =  e  —  v,  whence 
angle  OBC  =  90°—  (e  -f  v\  and  angle  OAC  =  90°—  (e  —  ,t>). 
The  angle  0/JB/S  =  90°  +  v,  and  angle  WA'A  =  90°  —  v. 


INACCESSIBLE    DISTANCES.  113 

In  the  triangle  O'B'B  we  have 

O'W      sin  [90°  —  (e  +  «)]  r'         cos  (e  + 1>) 

= or, = (a) 

O'B  sin  (90°  + t>)  WB  cos  D 

In  the  triangle  O'A'A  we  have 

O'A'      sin  [90°  —  (e  —  v)]  r'         cos  (e  —  v) 

= or, = (ft) 

O*A  sin  (90°  —  v)  2O'A  cos  v 

Adding  (a)  and  (6),  we  obtain 

r'r 

=  2  cos  e    (c). 

20' J?  X  O'A 

Multiplying  (a)  and  (6)  together,  we  obtain 

r'  r'  cos2  e  cos2  «  —  sin2  e  sin2  -e 

= (d) 

4  0'.B  X  O'A  cos2  •» 

Dividing  (c)  by  (d),  we  have,  after  a  little  reduction, 
r  cos  e 

-  =  - — ,   00 

r7      cos2  e  —  sin2  e  tan2  0 

tfhich  is  an  expression  of  the  relation  sought. 

Cor.— With  the  wires  adjusted  to  one  foot  on  the  rod 
for  a  base  of  100  feet,  we  should  have 

tan  v  =  0.005  ft.,  or  tan2  v  =  0.000025  ft 
Thus,  tan2  v  =  0,  without  material  error. 
Whence  formula  (e)  becomes  r'  =  r  cos  e. 

To  find  the  distance  CO'  we  have 

r' 
CO'  =  d'  =/-  +/+  c  =  Z/  +  /+  a. 

s 
Whence,  CO  =  d  =  (V  +  /  +  c)  cos  e. 

For  vertical  rod  we  have,  b'  »  6  cos  e. 
Whence,  d  =  6  cos2  e  +  (/  -f-  c)  cos  e.    (/) 

The  height  00' =  h  =  ±b  sin  2e  +  (/  +  c)  sin  e.    &) 

Example.— Given  e  =  10°  3CK,  r  =  5.36  ft.,  and  /  +  c  = 
1  ft.,  to  find  d  and  h. 

Solution. — Suppose  the  wires  adjusted  to  give  1  ft.  on 
the  rod  to  the  100  ft.,  whence  6  =  536  ft.  8 


114 


A  MANUAL   OF  LAND   SURVEYING. 


Cos  e  =  0.983  and  cos2  e  =  0.9668. 
Whence,  d  =  536  X  0.9668  -f  0.98  =  519.18  ft. 
Sin  e  =  0.182,  and  |  sin  2e  =  0.1792. 
Whence,  h  =  536  X  0.1792  +  0.18  =  96.23  ft. 

Formula  (/)  may  be  put  in  the  form 

d  =  I  cos2  e  -f  (/+  c)  cos2  e  +  (/+  c)  cos  e  (1  —  cos  e). 

Dropping  the  last  term,  we  have 

d  =  (6  -f /  -f  c)  cos2  e.    (7i) 

Assuming  ^ -f  c  =  1  ft.  as  a  mean  value  in  different 
instruments,  the  omission  of  the  term  (/  4-  c)  cos  e 
(1  —  cos  e)  introduces  an  error  for  ordinary  elevations  of 
less  than  0.01  ft.  in  a  base  of  1000  ft. 

Moreover,  the  use  of  formula  (h)  operates  to  diminish 
the  very  minute  error  introduced  by  use  of  formula  (/) 

For  slight  elevations,  as  from  1°  to  2°,  the  reduction  to 
horizontal  may  be  omitted.  For  5°  44X  the  amount  of  the 
reduction  is  about  one  per  cent.  The  correction  for  hori- 
zontal measurement  is  sometimes  made  by  omitting  to 
add/-f  c  to  the  base. 


INACCESSIBLE   DISTANCES.  115 

4.  The  Gradienter  is  an  attachment  to  the  transit 
for  fixing  grades  and  determining  distances. 

As  made  by  Gurley,  it  consists  of  a  screw  attached  to 
the  semicircular  expanded  arm  of  the  ordinary  clamp  of 
the  telescope  axis  ;  the  screw  is  accurately  cut  to  a  given 
number  of  threads,  and  passing  through  a  nut  in  one  side 
of  the  arm,  presses  against  a  little  stud,  A,  fixed  to  the 
inside  surface  of  the  right-hand  standard. 

In  the  other  side  of  the  semicircular  arm  is  inserted  a 
hollow  cylinder  containing  a  pin  actuated  by  a  strong 
spiral  spring,  the  end  of  the  pin  pressing  against  the  side 
of  the  stud  opposite  that  in  contact  with  the  screw. 

Near  the  other  end  of  the  screw,  and  turning  with  it, 
.is  a  wheel,  or  micrometer,  the  rim  of  which  is  plated  with 
silver,  and  divided  into  100  equal  parts. 

A  small  silver  scale,  attached  to  the  arm  and  just  above 
the  micrometer  wheel,  is  divided  into  spaces,  each  of 
which  is  just  equal  to  one  revolution  of  the  screw ;  so 
that  by  comparing  the  edge  of  the  wheel  with  the  di- 
visions of  the  scale,  the  number  of  complete  revolutions 
of  the  screw  can  be  easily  counted. 

It  will  be  seen  that  when  the  clamp  is  made  fast  to  the 
*ixis  of  the  clamp-screw,  and  the  gradienter-screw  turned, 
it  will  move  the  telescope  vertically,  precisely  like  the 
tangent-screw  ordinarily  used. 

And  as  the  value  of  a  thread  is  such  that  a  complete 
revolution  of  the  screw  will  move  the  horizontal  cross- 
wire  of  the  telescope  over  a  space  of  one  foot  on  a  rod 
at  a  distance  of  one  hundred  feet,  it  is  clear  that  when 
the  screw  is  turned  through  fifty  spaces  on  the  graduated 
head,  the  wire  will  pass  over  fifty  one-hundredth s,  or 
one-half  a  foot  on  the  rod,  and  so  on  in  the  same  propor- 
tion. 

In  this  way,  the  gradienter  can  be  used  in  the  measure 
ment  of  distances,  precisely  like  the  stadia. 

8 


116  A  MANUAL   OF   LAND   SURVEYING. 

Grades  can  also  be  established  with  great  facility,  as 
follows:  Level  the  instrument;  bring  the  telescope  level 
to  its  centre  by  the  clamp  and  gradienter  screw ;  move 
the  graduated  head  until  its  zero  is  brought  to  the  edge 
of  the  scale,  and  then  turn  off  as  many  spaces  on  the 
head  as  there  are  hundredths  of  feet  to  the  hundred  in 
the  grade  to  be  established. 

Having  a  transit  with  gradienter  attachment,  let  the 
student  solve  the  following  problems  in  the  field : 

Prob.  1.  To  find  the  grade  between  two  points. 

SUGGESTIONS.  — Set  the  instrument  over  one  of  the 
points,  level  the  plates  and  the  telescope,  and  bring  the 
zero  of  the  screw  to  the  edge  of  the  scale. 

Set  the  target  of  the  leveling  rod  at  height  of  instru- 
ment. 

With  the  rod  held  upon  the  other  point,  note  the  num- 
ber of  revolutions  of  the  screw  required  in  bringing  the 
cross- wire  upon  the  center  of  the  target.  That  number, 
as  so  many  feet,  is  the  grade. 

Prob.  2.  To  find  the  distance  between  two  points. 

SUGGESTIONS. — Set  up  and  adjust  the  parts  of  the  in- 
strument as  in  Prob.  1.  On  a  leveling  rod  held  upon  the 
other  point,  note  the  number  of  feet  covered  by  one  revo- 
lution of  the  screw,  and  multiply  that  number  by  100. 

If,  in  order  to  cover  r  feet  on  a  rod  at  a  distance  of  d 
feet,  n  revolutions  of  the  screw  are  required,  then  we 
should  have:  d  :  100  ::  r  :  n;  whence  d  =  100 r-s-  n. 

Example—  Given  n  =  2.30  and  r  =  5  ft.,  to  find  d. 

Kesult,  d  =  217.39  ft. 

On  inclined  ground  the  horizontal  sight  line  may  be. 
above  or  below  the  rod.  In  such  cases,  as  in  stadia 
measurement,  a  formula  of  reduction  to  a  horizontal  is 
employed,  which  may  be  deduced  as  follows: 


INACCESSIBLE   DISTANCES.  117 

Let  CO  —  d  (Fig.  34),  be  a  horizontal  sight  line; 

Angle  OC(y  =  e,  the  elevation  of  telescope  to  foot  of 
rod; 

Angle  O'CB  =  v,  the  angle  described  by  n  revolutions 
of  the  screw; 

O'W  =  r',  the  space  on  a  rod  perpendicular  to  CO*, 
subtending  angle  v,  and 

O'B  =  r,  the  corresponding  space  on  a  vertical  rod. 

We  shall  then  have,  [Formula  (a)\ 
r'      sin  [90°  —  (e  +  v)]      cos  e  cos  v  —  sin  e  sin  v 

r  sin  (90°  +  v)  cos  v 

Whence,  r'  =  r  (cos  e  —  sin  e  tan  «). 

r'        n 

Let  CO'  =  d'.    Then,  tan  v  =  —  =  — . 

d'      100 
100  r'       100  r  ( 

Whence,  d'  = = ]  cos  e  —  sin  e  X 

n  n     ( 


-1 

100) 

( 100  cos  e  V 

=  r} sineL 

(       n  ) 


or    d'  =  rj sine^.    (1) 

Now,  d  =  df  cos  e. 

(100  ) 

Whence,  d  =  r  ]  —  cos2  e  —  %  sin  2e  [  .    (2) 

(  n  ) 

Cor. — If  n  =  1,  we  have, 

d'=r  (100  cos  e  —  sin  e\         (3) 
and  d  =  r  (100  cos2  e  —  \  sin  2e),    (4) 
in  which  r  is  the  space  on  a  vertical  rod  included  by  one 
revolution  of  the  screw. 

The  numbers  by  which  this  value  of  r  must  be  thus 
multiplied  for  various  elevations  are  given  in  Table  IX. 

Examples.— 1.  Given  e  =  15°  20',  and  r  =  5.42  for  one 
revolution  of  the  screw,  to  find  d'  and  d. 

SOLUTION.— We  find  in  Table  IX, 

factor  for  inclined  distance  for  15°  =  96.33 
«•         "         "  "        15°  30'  =  96.09 


Difference  for  30'  =  0.24 
whence,         "  "  2(X=  0.16 


118  A  MANUAL   OF  LAND   SURVEYING. 

Whence,  factor  for  inclined  distance  for  15°  20X  =  96.17. 
Accordingly,  d  =  5.42  X  96.17  =  521.24  ft. 

Again,  in  Table  IX  we  have 

factor  for  horizontal  distance  for  15°  =  93.05 

"        "  "  "         15°  30'  =  92.59 

Difference  for  30/  =   0.46 
whence,  "   20'  =   0.31 

Whence,  factor  for  horizontal  dist.  for  15°  20'  =  92.74. 

Hence,  d  =  5.42  X  92.74  —  502.65  ft. 

2.  Given  e  '=  10.35  rev.  to  foot  of  rod,  and  r  =  6.25,  to 
find  d/  and  d. 

SUGGESTION.— From  Table  X  find  the  angle  e,  and  solve 
as  above. 

When  c  is  an  angle  of  depression,  the  point  (/  is  the  upper  end  of 
the  rod.  The  application  of  the  formula  is,  however,  the  same  in  this 
case  as  in  the  one  considered. 

Stadia  and    Gradienter    Measurements  are 

found  very  convenient  in  solving  some  of  the  problems 
in  land  surveying,  but  are  almost  useless  in  others.  They 
save  time  and  trouble  in  measuring  across  streams,  bogs 
and  other  places  inaccessible  to  the  chain  or  tape.  They 
furnish  a  quick  and  easy  means  of  determining  how  far 
it  is  to  an  object,  but  a  slow  one  of  locating  points  at  any 
desired  distance,  such  as  setting  stakes  for  a  town  plat,  a 
ditch  line,  or  a  railroad. 


PLATTING. 


CHAPTER  VI. 
PLATTING  AND  COMPUTING  AREAS. 

1.  A  Plat  or  Plot  is  a  representation,  upon  a  small 
scale,  of  the  lines  of  a  survey.  Platting  is  simply  sur- 
veying on  paper.  The  instruments  used  are  analogous  to 
those  used  in  the  field. 

Lines  are  marked  upon  the  paper  with  pencil  or  pen 
and  ink.  Generally  they  will  first  be  drawn  lightly  in 
pencil;  afterward  the  permanent  lines  will  be  inked,  and 
all  erroneous  or  superfluous  lines  erased.  Pencils  hard 
enough  to  hold  a  fine  point  without  breaking  are  the  best 
for  this  use. 

The  right  line  pen  is  used  for  drawing  straight  lines, 
it  is  made  in  various  sizes  and  forms.  One  of  the  best  is 
shown  at  6,  in  Figure  36. 

The  scale  of  equal  parts  is  the  counterpart  of  the  chain 
or  tape.  A  great  variety  of  scales  are  made.  One  of  the 
most  useful  is  the  triangular  scale  (Fig.  36,  e).  It  has  six 
different  graduations,  all  brought  to  the  edge,  so  that  the 
scale  may  be  laid  down  on  the  paper  and  the  distance 
marked  off  directly  from  the,  scale.  The  scale  in  which 
the  inch  is  divided  into  10,  20,«30,  40,  50  and  60  equal  parts 
is  the  one  most  useful  to  the  surveyor.  Paper  scales  are 
made  on  fine  Bristol  board,  with  any  graduation  desired. 
They  are  cheap,  and  as  good  as  any  scale  as  long  as  they 
last.  The  student  may  make  his  own  scales  on  paper. 

The  protractor  (Fig.  36,  a)  takes  the  place  of  the  com- 
pass or  transit.  It  is  simply  the  whole  or  part  of  a  grad- 
uated circle  or  limb.  Protractors  are  made  in  a  great 
variety  of  forms.  One  of  the  cheapest  and  best  has  the 


120  A  MANUAL  OP  LAND  SURVEYING. 


PLATTING.  121 

entire  circle  graduated  to  quarter  degrees.  It  is  made  of 
paper,  has  the  middle  part  cut  out,  and  fine  threads  or 
wires  crossing  at  the  centre  of  the  circle.  A  paper 
protractor  14  inches  in  diameter,  graduated  to  quarter 
degrees,  costs  from  30  to  40  cents. 

Dividers,  (Fig.  36, /)  are  used  to  space  off  distances  on 
the  plat,  or  transfer  distances  from  the  scale  to  the  plat 
or  the  reverse.  When  provided  with  pen  or  pencil  points 
they  are  used  to  strike  circles  and  arcs.  When  they  are 
used  for  the  latter  purpose  they  should  have  a  needle 
point  on  the  stationary  leg. 

Parallel  rulers,  as  the  name  indicates,  are  used  in 
drawing  parallel  lines.  When  a  paper  protractor  is  used 
in  platting,  it  is  found  convenient  to  fasten  it  at  some 
point  outside  the  plat  and  transfer  the  bearing  of  the 
lines  from  the  protractor  to  the  plat  by  means  of  the 
parallel  rule.  The  best  rule  for  this  purpose  moves  upon 
rollers,  (Fig.  36,  d) 

The  straight-edge  ruler  and  triangle  are  also  used  to 
mark  parallel  lines,  as  well  as  to  lay  off  angles.  Many 
other  articles  will  be  found  convenient  in  platting.  A 
drawing  board,  made  of  the  softest  wood,  planed  smooth 
and  true,  and  thumb-tacks  to  fasten  the  paper  to  the 
board,  may  almost  be  considered  as  necessaries.  Neither 
the  student  nor  surveyor  needs  many  instruments  for 
platting,  but  those  he  has  should  be  perfect  in  their  kind. 
It  is  not  deemed  necessary  at  this  point  to  give  further 
details  of  these  instruments  and  their  uses,  any  sugges- 
tion which  the  student  may  need  being  left  to  the  teacher 
to  make. 

EXERCISES. 

The  first  seven  exercises  are  the  elementary  problems  of  Geometry, 
and  are  designed  to  be  solved  on  paper  by  use  of  the  dividers  and 
ruler. 

2.   1.  To  draw  a  straight  line  equal  to  a  given  straight 
line. 
2.  To  make  an  angle  equal  to  a  given  angle. 


122  A    MANUAL  OF  LAND  SURVEYING. 

3.  To  draw  through  a  given  point  a  line  parallel  to  a 
given  line. 

4.  To  draw  through  a  given  point  a  line  perpendicular 
to  a  given  line.    Two  cases. 

5.  To  bisect  a  given  line;  a  given  angle. 

6.  To  construct  lines  proportional  to  given  lines. 

7.  To  construct  a  polygon  similar  to  a  given  polygon. 

8.  Plat  the  following  lines : 

(1)  8  chains,  to  scale  of  2  chains  to  the  inch. 

(2)  10  chains,  to  scale  of  5  chains  to  the  inch. 

(3)  10  chains,  to  scale  of  4  chains  to  the  inch. 

(4)  17.25  chains,  to  scale  of  3  chains  to  the  inch. 

(5)  25.40  chains,  to  scale  of  4  chains  to  the  inch. 

9.  Plat  a  triangle  whose  sides  are  13.50  eh.,  14.25  ch.  and 
16.20  ch.,  on  a  scale  of  5  chains  to  an  inch;  on  a  scale  of  3 
chains  to  an  inch. 

10.  Plat  a  rectangle  whose  adjacent  sides  are  9.24  ch. 
and  13.78  ch.,  on  a  scale  of  4  chains  to  the  inch. 

11.  Plat  a  quadrilateral  the  sides  of  which  are  22.60  ch., 
14.35  ch.,  12.20  ch.  and  9.80  ch.,  on  a  scale  of  4  chains  to 
the  inch,  and  having  one  angle  of  83°  30'. 

12.  Measure  the  remaining  angles  and  find  their  sum. 

13.  Plat  any  figure  having  five  equal  sides;  measure  the 
interior  angles  and  find  their  sum. 

14.  Plat  a  right  triangle  having  a  base  of  16.25  ch.  and 
a  perpendicular  of  8.60  ch.    Find  the  remaining  side  and 
angles  of  the  triangle. 

II.     COMPUTING  AREAS. 

In  land  surveying  the  areas  are  computed  in  triangles 
and  quadrangles.  If  a  field  has  more  than  four  sides,  in 
making  the  computation  it  is  parted  off  into  triangles 
and  rectangles  or  trapezoids,  the  area  of  which  is  com- 
puted and  their  sum  taken. 

1.  Area  of  Triangles. 

1.  To  find  the  area  of  a  right  angled  triangle. 
Multiply  the  base  by  one  half  the  perpendicular. 


COMPUTING   AREAS.  123 

2.  To  find  the  area  of  an  oblique  angled  triangle. 
CASE  IST. —  When  the  sides  are  given. 

Let  A,  B,  C  represent  the 
angles,  and  a,  6,  c  the  sides 
opposite  them. 

a-f  6-f-  c 

Let =  s.  Let  x = area. 

FIG.  37.  2 

Then  x  =  ^/X*  —  «)  (*— 6)  0— °)« 
CASE  2^D.— Having  two  sides  and  the  included  angle 

Let  a,  6  be  the  sides.  C  the  given  angle,  and  x  =  area. 

From  J?  drop  a  perpendicular,  d,  to  the  side  6.    This 

divides  the  triangle  into  two  right  triangles,  the  area  of 

each  of  which  equals  its  base  multiplied  by  half  the 

perpendicular,  d,  and  the  sum  of  their  areas  equals  the 

sum  of  their  bases  multiplied  by  half  the  perpendicular; 

bd  ab  sin  C 

that  is,  x  —  — .    But  d  =  a  sin  C.    Hence,  x  = . 

2  2 

CASE  3D.— Given  two  angles  and  the  included  side. 

Let  A  and  B  be  the  angles,  and  c  the  side  given, 
Find  C  =  180°  —  (A  -f  S).    Find  6. 

c  sin  .B  60  sin  J. 

Sin  C  :  sin  B  : :  c  :  6     :.  b  = x  = . 

sin  C  2 

CASE  4TH.— Given  two  angles  and  a  side  opposite,  (A,  B 

and  a.) 

a  sin  C 

Find  C  =  180°  —  (4  -f  -B).    Find  c  = . 

sin  A 

a  sin  .5  6c  sin  A 

Find  6  = .    Then  x  = . 

sin^L  2 

2.  Areas  of  Quadrangles. 

CASE  IST.— Squares  and  rectangles. 
Multiply  the  base  by  the  perpendicular. 


A  MANUAL  OF   LAND   SURVEYING. 


CASE  ZNV.—Trapezoids.  A  trapezoid  is  a  figure  having 
four  sides,  only  two  of  which  are  parallel. 

Its  area  is  equal  to  the  half 
sum  of  the  parallel  sides,  multi- 
plied by  the  perpendicular  dis- 
tance between  them. 
Trapezoid.    FIG.  38. 

CASE  3RD. — Trapeziums  have  no  two  sides  parallel. 

The  area  is  found  by  parting 
off  into  triangles  and  comput- 
ing their  areas. 

1.    Having   the   sides   and 
Trapezium.    FIG.  39.          angles  given. 

Let  A,  B,  C,  D  represent  the  angles,  and  a,  6,  c,  d  the 

sides  of  the  trapezium.    Let  AC  be  a  diagonal  dividing 

the  trapezium  into  the  triangles  ABC  and  ADC.    In  each 

of  these  we  have  two  sides  and  an  included  angle  given; 

ab  sin  B      cd  sin  D 

hence,  x  = 1 . 

2  2 

2.  Given  the  diagonals  of  a  quadrilateral  and  an  angle 
formed  by  their  intersection,  to  find  the  area. 

Solution.— Let  ABCD  be  the 
quadrilateral,  m  and  n  its 
diagonals,  and  0  an  angle  at 

which  the  diagonals  intersect. 

FIG.  40. 

By  Case  2nd,  under  "Area  of  Triangles," 
area  AOB=  \AOXBO  sin  0 


"     DOC  =  1  CO  X  DO  sin  0 
"     BOG  =  1  CO  X  SO  sin  0. 

Whence,  by  addition,  area  ABCD  —  \  (AO  +  CO)  X 
(BO  +  DO)  sin  0, 

mn  sin  0 
or,  area  ABCD  = 


COMPUTING  AREAS.  125 

Example.—  The  diagonals  of  a  four-sided  field  were 
found  to  measure  18  ch.  and  24  ch.  Setting  a  compass  at 
their  intersection,  the  bearings  of  two  adjacent  corners 
of  the  field  were  found  to  be  N.  30£°  E.  and  S.  50°  E. 
Required  the  area  of  the  field. 

Solution.—  Applying  logarithms  in  the  above  formula, 
having  found  0  =  99|°,  we  have 

«i  =  18  log         1.255273 

n  =  24  log         1.380211 

0  =  99£°  log  sin  9.994003 

2    ar.  co.  log        9.698970 

Area  =  213.03  log        2.32S457 

or,  Area  =  21.303  A. 

5.  Given  three  sides,  a,  6,  d,  and  the  included  angles,  A 
*nd  D.  (See  Fig.  39.) 

Let  AC  =  e,  be  a  diagonal.  Let  the  angle  BCA  =  E, 
BAG  =  F,  and  CAD=G.  In  the  triangle  ABC  the  sides 
a,  b  and  angle  B  are  known.  In  the  triangle  CAD  the 
side  d  only  is  known.  It  is  required  to  find  the  side  e  and 
the  angle  G.  To  find  G  :  E  +  F  =  180°  —  B.  By  tri  go- 

tan  E  -f  F      tan  E  —  F 

nometry,  a  +  6  :  a  —  b  ::  -  :  -  ,  by 

2  2 

which  we  find  the  sum  and  the  difference  of  the  angles  E 

E+F      E—F 

and  F.     ----  =  F,  and  G  =  A  —  F. 
2  2 

6  sin  B 
To  find  e  :  Sin  F  :  sin  B  :  :  b  :  e     :.  e  —  -  —  . 


4.  This  method  of  finding  the  area  of  a  trapezium  may 
be  applied  to  polygons  of  any  number  of  sides,  when  the 
sides  and  angles  are  given.  The  polygon  is  divided  into 
triangles  two  less  in  number  than  the  number  of  sides 
Each  triangle  has  two  sides  and  the  included  angle  given 
or  readily  found. 


126  A   MANUAL   OF   LAND   SURVEYING. 

Take  for  example  the  irregular  polygon  of  eleven  sides 
shown  in  Fig.  41,  which  is  divided  into  nine  triangles. 

In  the  triangles  A,  B,  C  and 
7)  two  sides  and  the  included 
angle  of  each  are  given.  From 
the  remaining  sides  and  angles 
we  find  two  sides  and  the  in 
eluded  angle  of  the  triangles 
E  and  F,  and  so  each  triangle 

in  turn  furnishes  the  data  for  computing  the  adjacent 

triangle,  till  all  are  complete. 

3.  Offsets.— When  it  is  desired  to  find  the  area  of 
a  field  having  irregular  sides,  such  as  along  a  stream  or 
lake,  it  is  well  to  run  a  straight  line  where  most  conve- 
nient to  do  so,  and  then  run  and  measure  perpendiculars 
to  the  margin  of  the  field.  These  are  called  offsets.  They 
divide  the  space  between  the  straight  line. and  the  margin 

of  the  field  into  triangles 
and  trapezoids,  whose 
areas  may  be  computed 
separately  and  the  sum 
taken. 


If  the  offsets  are  equidistant  the  area  may  be  found  by 
the  following 

RULE. — From,  the  sum  of  the  offsets,  subtract  the  half 
sum  of  the  extreme  ones,  and  multiply  the  remainder  by 
the  common  distance  between  them. 

4.  What  is  the  area  in  acres  of  the  following  rigM 
angled  triangles? 

1.  Base  =  23.20  ch.,  perpendicular  =  14.60  eh.? 

Ans.  16.936  .4 

2.  Base  =  19.46  ch.,  perpendicular  =  12.18  ch.  ? 

What  is  the  area,  in  acres  of  the  following  oblique 
angled  triangles :  (See  Fig  37.) 

3.  a  =  14.26  ch.,  6  =  19.40  ch.,  c  =  12.18  ch.  ?  Ans.  8.666  A 

4.  a=  9.43    «    6  =  11.61    "    c=   8.42  " 


COMPUTING   AREAS.  127 

5.  a=  6.23    "    6  =  14.26    "   (7  =  22°  40'?   Am.  1.11+ A. 
6.a  =  12.20   "    6  =  20.00    "  C=36°  15'? 

7.  ^  =  16045',  .B  =  82°30',  c  =  21.16ch.?  Am.  6.458+  A. 

8.  J.  =  35°,        #  =  62°  42',  c  =  18.20   " 

9.  ^1  =  46°,        5  =  58°  15',  a  =  26^50   "       An*.  40.264  4. 
10.  A  =  37°  20',  B  =  72°  40',  a  =  19.36   tt 

11.  A  square  field  is  6.25  chains  on  a  side.    Required  its 
area. 

12.  A  square  field  contains  20  acres.   What  is  the  length 
of  its  sides?  Ans.  14.142  ch. 

13.  What  is  the  area  of  a  rectangle  whose  sides  are 
16.41  and  8.26  chains  ? 

14.  A  rectangular  field  containing  16  acres  measures 
12.50  chains  on  the  base.    What  is  the  perpendicular  ? 

Ans.  12.80  ch. 

15.  Commencing  on  the  margin  of  a  river  a  line  was 
run  across  a  bend  20.00  chains  to  the  margin.    Commenc- 
ing at  the  end  of  the  second  chain,  offsets  were  taken 
every  two  chains,  to  the  margin  of  the  river,  as  follows: 
1.61  ch.,  2.27  ch.,  3.72  ch.,  1.96  ch.,  4.23  ch.,  2.92  ch.?  3.26  ch., 
2.50  ch.  and  1.25  ch.    Required  the  area  between  the  line 
and  the  river.  Ans.  4.744  acres. 

16.  Required  the  area  of  a  field  bounded  as  follows: 
1st.    North  17.65  ch. 

2nd.  8.  36°  12/  W.  8.20   ch. 
3rd.   S.  12°  34'  W.  7.26     " 
4th.  S.  58°  26'  E.  7.53^  " 

SUGGESTION.  —  First :  Change  bearings  into  angles 
between  the  lines  and  compute  as  two  triangles. 

Second:  Take  the  first  line  as  a  base,  divide  the  figure 
into  two  right  angled  triangles  and  a  trapezoid,  and  com- 
pute the  area.  Compare  the  two  methods  as  to  number 
of  figures  required  for  the  solution. 

17.  The  sides  of  a  pentagon  measure  6.25  chains  each. 
What  is  its  area  ? 

SUGGESTION.— Part  the  figure  into  three  triangles  and 
compute.  Also  part  into  five  isosceles  triangles.  Com- 
pute and  compare  the  two  methods. 


128  A   MANUAL   OF   LAND   SURVEYING. 


f 


5.  1.  Rectangular  Coordinates.  —  Let  XX'  and 

YY'  be  two  lines  intersect- 
ing each  other  at  right 
angles,  as  at  0. 

Let  P:,  P2,  P3  be  any 
p          points  in  the  plane  of  the 
j  I  lines. 

j  LetP^,  P2a2»  PaQa  De 

— =; "-or — ^  perpendiculars    from    the 

points  upon  the  axis  XX' , 
FIG.  43.  and  P^,   P262,  P363    be 

perpendiculars  from  the  points  upon  the  axis  YY'. 

The  distances  Oa:i  0«2,  Oa3  are  called  Abscissas  of 
the  points  Plt  F?,  P3;  and  the  distances  Ob^  Ob2,  Ob3  are 
called  Ordi nates  of  the  points. 

The  point  0  is  called  the  Origin. 

The  abscissa  and  ordinate  of  a  point  are  together  called 
Coordinates  of  the  points. 

Coordinates  at  right  angles  with  each  other  are  called 
Rectangular  Coordinates. 

It  is  customary  to  denote  abscissas  by  x  and  ordinates 
by  y,  coordinates  of  different  points  in  connection  with 
each  other  being  distinguished  by  use  of  subscripts. 

Thus,  of  the  point  Plt  the  coordinates  Oa-^  and  06j  or 
a1Pl  may  be  denoted  by  tf;t  and  yl ;  of  the  point  P2,  the 
coordinates  Oa2  and  062  or  a2P2  may  be  denoted  by  x2 
and2/2;  and  so  on. 

It  will  be  seen  that  the  coordinates  of  a  point  afford 
the  means  of  locating  it  with  respect  to  the  axes. 

The  use  of  longitude  and  latitude  in  Geography  is  an  illustration. 

By  use  of  the  signs  4-  and  — ,  the  coordinates  of  any 
point  in  the  plane  of  the  axes  are  readily  expressed. 


COMPUTING   AREAS.  129. 

EXERCISES. 

2.—  1.  Construct  the  point  of  which  x  =  4  and  y  =  7. 

2.  Given  #  =  —  5  and  y  =  3,  to  construct  the  point. 

3.  Given  x  =  —  3  and  y  =  —  6,  to  construct  the 

point. 

4.  Given  x  =  6  and  y  =  —  4,  to  construct  the  point 

5.  Given  a-  =  0,  y  =  2;  or  =  —  5,  y  =  0;  x  =  0,  y  =  0. 

Required  the  points, 

3.  Application  to  Area.  —  Let  it  be  required  to 
find  the  area  of  a  series  of  trapezoids  included  between 
perpendiculars  from  the  points  of  a  broken  line  upon  a 

straight  line.  Suppose 
the  straight  line,  as  0J7, 
to  be  an  axis  of  abscis- 
sas, and  the  first  perpen- 
dicular at  the  left,  as  OA, 

to  be  an  axis  of  ordinates. 
FIG.  44. 

Let  xitx2,xa,  etc.,  be  the  abscissas  of  the  points  A,  £t 
C,  etc.,  and  yi,y2,ya,  etc.,  the  corresponding  ordinates. 

Accordingly,  the  area  of  the  several  trapezoids  is 
\  [»a  (2/i  +  #2)  +  (#3  —  a?a)  fa*  +  #3) 

+  (a?4  —  a?8)  (2/3  +  y*)  H  ----  (av,  —  #n-i)  &n-i 
in  which  n  is  the  number  of  trapezoids  plus  one. 

The  above  formula  may  be  changed  to  the  form 


Whence,  for  the  area  included  between  a  straight  line, 
as  a  base,  and  a  broken  line  whose  points  are  given  by 
their  coordinates  upon  the  base,  we  have  the  following 

RULE.  —  From  each  ordinate  subtract  the  second  site- 
ceeding  one  and  multiply  the  remainder  by  the  abscissa 
corresponding  to  the  intervening  ordinate. 

Also,  multiply  the  sum  of  the  last  two  ordinates  by  the 
last  abscissa. 

Divide  the  algebraic  sum  of  the  products  by  2. 


130 


A   MANUAL   OF   LAND   SURVEYING. 


The  above  formula  and  Rule  have  been  deduced  independently  of 
any  supposition  as  to  the  relative  directions  of  the  parts  of  the  broken 
line.  They  are  therefore  true  whatever  may  be  the  form  of  the  broken 
line.  That  is,  whether  any  part  should  be  perpendicular  to  the-base, 
either  toward  or  from  it,  or  whether  any  part  should  be  turned  back- 
ward respecting  the  preceding  one. 

SUGGESTION.— Let  the  student  verify  the  rule  in  a  case, 

for  example,  like  the 
following,  in  which  BC 
is  represented  as  being 
parallel  to  the  base,  CD 
as  perpendicular  toward 
it,  and  FG  as  being 
I  turned  backward  from 
EF. 

Find  how  it  would  be,  if  one  or  more  of  the  ordinates 
were  zero;  if  one  or  more  were  negative. 


4.—  1.  Given  y 
also  xl  =  10,  x2 
area. 


EXERCISES. 

-  12,  */2  =  12,  yz  =  16,  */4  =  8  and  7/5  =  6t 
18,  #3  =  24,  x±  =  30  and  x5  =  20,  to  find 


Given  the  following,  to  find  area : 


(2) 


(3) 


(4) 


140 
435 


250 
200 


320 


1000 

000 

950 

812 

240 

844 

725 

306 

530 

500 

G40 

325 

450 

415 

200 

000 

000 

000 

1000 
1150 


828 
650 


460 
000 


200 
317 


420 
305 


524 
250 


5.  As  a  second  example  of  the  application  of  coor- 
dinates in  finding  area,  let  there  be  taken  an  ordinary 
polygon,  as  APCDEF.  (Fig.  46.) 

Let  a?!,  x2>  X3*  e^c.,  be  the  abscissas  of  the  points  A,  B, 
C,  etc,,  and  ylt  yz,  y3,  etc.,  tae  corresponding  ordinates. 


COMPUTING  AREAS. 


131 


0 


<t 


FIG.  46. 


Now,  since  formula  (a)  is  true  for  any  broken  line,  it 
holds  for  the  case  in  which  the  broken  line  beginning,  as 
at  A,  returns  to  the  same  point,  forming  thus  a  polygon, 
as  ABCDEFA. 

In  this  case,  the  last  term  of  (a)  vanishes,  and  we  have 
as  the  area  a  polygon  of  n  sides, 

\  [^i  (Vu—  2/2)  +  a?a  (#1—  2/3)  +  3?3  (2/2—  2/4)  -f  #4  (#3—  2/5)" 
+  etc.,  to  n  terms].  (6) 

or,  factoring  with  respect  to  y,  we  have  the  form 

—  s[2/i  (#•»—  arz)  +  2/2  Oa~  a;3)  +  ^3  (#2—^4)  -h  2/4  (^3—^5) 
+  etc.,  to  n  terms].  (c) 

Whence,  for  the  area  of  a  polygon  whose  vertices  are 
given  by  their  coordinates,  we  have  the  following 

RULE. — From  the  ordinate  of  each  vertex  subtract  the 
second  succeeding  one,  and  multiply  the  remainder  by  the 
abscissa  of  the  intervening  vertex;  or,  from  the  abscissa 
of  each  vertex  subtract  the  second  succeeding  one,  and 
multiply  the  remainder  by  the  ordinate  of  the  intervening 
vertex. 

Divide  the  sum  of  the  products  by  2 


132 


A  MANUAL   OF  LAND   SURVEYING. 


SCH.— Formulas  (b)  and  (e)  will  be  seen  to  be  in  accordance  with 
any  situation  of  the  coordinate  axes,  agreeably  with  convenience  of 
field  work.  In  particular  cases,  one  or  more  terms  will  be  found  to 
disappear.  Due  attention  to  algebraic  signs  is  important. 

The  formulas  are  easy  to  remember,  and  simple  of  application. 
With  an  instrument  adapted  to  lading  off  right  angles,  they  afford  a 
practical  means  of  computing  the  contents  of  irregular  tracts. 

EXERCISES. 

6.  Required  the  area  and  a  plat  of  a  field  the  coordi- 
nates of  whose  corners  are 

3^=3^=0,  xl=l  ch.,  »2=12|  eh.,  x3=lS  ch.,  a4=15  ch., 
#6  =  10  ch.;  and 

2/n  =  2/6  =  6  ch.,  2/1==12  ch.,  y2  =  20  ch.,  y8  =  15  ch., 
2/4=  8£  ch.,  2/5  =  0  ch.  Area,  16.175  acres. 

Find  the  area,  supposing  a  different  situation  of  the 
axes. 

7.  Given  the  lengths  and  bearings  of  the  sides  of  a 
polygonal  field,  to  find  the  area. 

Solution.  —  Let 
AS  CDS  represent 
the  field.  Let  NS  de- 
note the  meridian  of 
the  most  westerly 
station.  This  line, 
which  may  be  as- 
sumed as  passing 
through  any  station 
at  pleasure,  but  more 
conveniently  the  ex- 
treme western  or  the 
eastern  one,  is  called 
the  Principal  Me- 
ridian. 

To  the  principal 
meridian  let  there  be 
drawn  from  the  sev- 
eral stations  the  per-  FIG.  47. 


COMPUTING   AREAS.  133 

pendiculars  Ba,  Cd,  Dh  and  Ek,  and  upon  Cd  and  Dh  let 
there  be  drawn  the  perpendiculars  Bb,  Cc  and  Ee. 

These  perpendiculars  are,  respectively,  -the  bases  and 
the  altitudes  of  trapezoids  composing  a  portion  of  the 
field. 

Now,  if  from  the  sum  of  the  areas  of  the  trapezoids 
the  sum  of  the  areas  of  the  triangles  ABa  and  AEk  be 
subtracted,  the  remainder  will  be  the  area  sought. 

That  is,  clearing  of  fractions, 
2  X  area  pol.  =  (aB  +  Cd)Bb  +  (dC  +  Dh)  Cc  +  (hD  +  EK) 


It  is  now  to  be  considered  how  the  dimensions  of  the 
tra'pezoids  and  triangles  depend  upon  the  lengths  and 
bearings  of  the  sides  of  the  field. 

8.  Latitude  and  Departure.—  For  convenience  of 
description,  let  it  be  supposed  that  a  survey  of  the  field 
above  represented  was  made  "  with  the  land  on  the  right,'* 
beginning  at  A. 

In  going  from  A  to  B,  there  was  made  a  distance  Aa, 
north,  and  a  distance  aB,  east;  in  going  from  B  to  C  there 
was  made  a  distance  Bb,  south,  and  a  distance  bC,  east. 
Finally,  in  going  from  E  to  A,  there  was  made  a  distance 
kAt  north,  and  a  distance  Ek,  west.  Distances  made  north 
are  called  Northings,  and  south,  Southings;  dis- 
tances made  east  are  called  Eastings,  and  west,  West- 
ings. Northings  and  southings  are  together  called 
Latitudes,  and  eastings  and  westings  are  called 
Departures. 

It  will  be  seen  that  the  length  of  a  course  is  the  hypot- 
enuse of  a  right  triangle  of  which  the  latitude  of  the 
course  is  the  side  adjacent  to  the  bearings,  and  the  depar- 
ture, the  side  opposite  the  bearing.  Whence, 

Latitude  =  length  of  course  X  cosine  of  bearing,  and 
Departure  =  length  of  course  X  sine  of  bearing. 

From  these  fundamental  formulas,  several  others  ex- 


134 


A  MANUAL   OF   LAND  SURVEYING 


pressing  relations  of  either  of  the  four  quantities  to  two 
others  are  easily  derived. 

Thus,  denoting  the  latitude  by  Z,  the  departure  by  d,  the 
length  of  course  by  c,  and  the  bearing  by  b,  is  obtained 
the  following 

TABLE    OF  CASES. 


No. 

Given. 

Required. 

Formulas. 

1 

fc,  c 

I,  d 

1  =  c  cos  6        d  —  G  sin  6. 

1     - 

2 

b,  I 

c,  d 

c  =  d  =  l  tan  6. 

cos  6 

r.    fi 

/>    7 

d                        d 

f                                7 

P,   (I 

C,    I 

sin  b                 tan  6 

Z 

4 

C,l 

6,  d 

cos  6  =  d  =  v"c2  —  I2. 

G 

d 

5 

c,d 

b,t 

sin  6  =  1  —  i/c2  —  d2. 

G 

d 

6 

I,  d 

0,  C 

tan  b  =  c  =  i/Z2  -f  d'2. 

I 

The  Traverse  Table.— This  table,  which  is  given 
with  others  in  the  back  part  of  the  book,  shows  the  lati- 
tude and  departure  for  any  bearing  to  each  quarter  degree 
for  any  distance  from  1  to  10.  For  other  distances,  the 
latitude  or  departure  is  found  by  adding  the  latitudes  or 
the  departures  of  the  partial  distances,  as  shown  in  the 
following 

EXERCISES- 

9.— 1.  Find  the  latitude  and  the  departure  for  a  bearing 
of  24°,  for  a  distance  of  7  ch.;  for  a  distance  of  5  eh.;  for 
a  distance  of  10  cK 


COMPUTING   AREAS.  135 

2.  Find  the  latitude  and  the  departure  on  a  bearing  of 
371°,  for  a  distance  of  12  ch. 

OPERATIONS. 

For  37i°,  distance  10,  lat.  =  7.9600  dep.  =  6.0529 
"       "  "  2,     "    =1.5920    "      =1.2106 


"       "                       12,    "    =9.5520;  "  =7.2635. 

3.  Find  the  latitude  and  departure  on  a  bearing  of  40f  ° 
for  a  distance  of  17.23  ch. 

OPERATIONS. 

For  40f, distance  10,    lat.  =   7.5756,    dep.=  6.5276 

"       "             "         7,      "<  ==   5.3030,       "  =  4.5693 

.2   "    ==  0.15151,     «  =  0.13055 

"       "            "           .03"    =  0.022727,  "  =  0.019583 

"      "            "       17.23"    =13.053,        "  =11.247 

ANOTHER   FORM   OF   WORK. 

Bearing.          Distances.                  Latitude*.  Departures. 

40|°                1000                   07576  06528 

700                    53030  45693 

20                      15151  13055 

3                       22727  19583 


1723  1305.3237  1124.7433 

"We  take  the  distance  in  links,  and  write  the  latitude  and  departure 
for  the  first  figure  of  the  number,  omitting  the  decimal  point ;  we  write 
under  them  the  latitude  and  departure  for  the  second  figure,  setting 
them  down  one  place  farther  toward  the  right ;  under  them,  the  lati- 
tude and  departure  for  the  third  figure,  setting  them  one  place  farther 
toward  the  right,  and  so  on. 

We  then  add  the  separate  latitudes  and  separate  departures,  and 
point  off  four  figures  from  the  right.  The  results  thus  obtained  are  the 
latitude  and  departure  sought,  as  expressed  in  links. 

Notice  that  bearings  from  45°  upward  are  found  in  the  right  hand 
column  of  the  table,  and  the  columns  of  latitude  and  departure  are 
denoted  at  the  foot  of  the  page.  Care  needs  to  be  taken  here  to  avoid 
mistakes  of  latitudes  for  departures  and  departures  for  latitudes. 


136  A   MANUAL    OF    LAND    SURVEYING. 

Find  the  latitudes  and  departures  for  the  following 
bearings  and  distances : 

(1)  Bearing  52|°,  Distance  437. 

(2)  Bearing  65£°,  Distance  3669. 

(3)  Bearing  21  f°,  Distance  2030. 

(4)  Bearing  40°,  Distance  506. 

(5)  Bearing  81|°,  Distance  12.34  ch. 

1O.  Meridian  Distance. — The  distance  of  a  station 
or  any  point  from  the  principal  meridian  is  called  its 
Meridian  Distance.  The  meridian  distance  of  a  line 
is  the  meridian  distance  of  its  middle  point.  If  the 
meridian  passing  through  the  extreme  easterly  or  west- 
erly station  of  a  survey  around  a  tract  of  land  be  taken 
as  a  base  and  perpendiculars  be  drawn  from  it  to  each 
station  of  the  survey,  the  tract  and  the  space  between, 
it  and  the  meridian  will  be  divided  into  triangles  and 
trapezoids  whose  areas  are  readily  computed. 

Beginning  with  the  station  through  which  the  me- 
ridian passes  which  we  call  Sta.  0,  then  the  meridian 
distance  of  Sta.  1  will  equal  the  departure  of  the  first 
course. 

The  meridian  distance  of  any  station  will  equal  the  alge- 
braic sum  of  the  departures  of  all  the  preceding  courses 
up  to  that  point. 

The  meridian  distance  of  any  course  or  line  will  equal  the 
half  sum  of  the  meridian  distances  of  the  stations  at 
the  two  ends  of  that  course  or  line. 

The  area  of  any  triangle  or  trapezoid  thus  formed  will 
equal  the  product  of  the  latitude  of  the  line  or  course 
on  which  it  is  based  multiplied  by  the- meridian  dis- 
tance of  that  line. 

The  area  of  the  tract  is  equal  to  the  sum  of  the  areas  of 
all  the  triangles  and  trapezoids  thus  formed  minus  the 
sum  of  the  areas  of  those  triangles  and  trapezoids  which 
lie  outside  the  lines  of  the  survey. 

The  area  of  the  tract  is  also  equal  to  the  difference 
between  the  sums  of  those  areas  found  from  latitudes 
which  are  northings  and  of  those  where  they  are 
southings. 


COMPUTING    AREAS. 


137 


We  will  now  apply  the  foregoing  principles  to  find 
the  area  of  the  tracts  described  in  the  following  Field 
Notes  and  shown  in  the  figure.  On  the  figure  each  sta- 
tion is  numbered  to  correspond  with  the  field  notes  and 
each  line  is  also  numbered  in  its  order  as  run.  The  sev- 
eral triangles  and  trapezoids  formed  by  perpendiculars 
from  the  stations  to  the  meridian  are  lettered  in  their 
order. 


Station 

Bearing 

Distance 

0 

N.  26i°  E. 

12.00  ch. 

1 

X.  59°  E. 

9.80  " 

2 

S.  66°  E. 

19.60  •/ 

3 

S.  35°  W. 

15.68  " 

4 

S.  66°  W. 

13.12  " 

5 

N.  46°  W.       11.72  H 

Bearing  26i° 

Dist.  12.M 

Lat.  10.74  N. 

"       59° 

9.80 

"      5.05  N. 

"       66° 

"      19.60 

«      7.97  S. 

"       35° 

"      15.68 

"    12.85  S. 

"       66° 

"      13.12 

"      5.34  S. 

"       46° 

"      14.72 

"    10.36  N. 

Finding  from  the  Traverse  Table  the  latitudes  and 
departures  to  the  nearest  link,  we  have 

Dep.  5.35  E. 

"     8.40  E. 

"    17.91  E. 

"     8.99  W. 

"    11.99  W. 

"    10.59  W. 

Obviously,  in  going  entirely  around  a  field  there  should 
be  made  the  same  southing  as  northing,  and  the  same 
westing  as  easting.  But  from  unavoidable  lack  of  pre- 
cision in  the  use  of  instruments,  this  is  practically  seldom 
found  to  have  been,  done,  according  to  the  figures  used. 
The  error,  however,  can  usually  be  made  very  small. 
Finding  it  large,  the  entire  field  work  should  be  reviewed. 

It  is  not  a  settled  point  among  surveyors  how  great  an  error  of  lati- 
tude or  departure  may  be  allowed  without  resurveying  the  lot.  Some 
would  admit  a  difference  of  one  link  for  every  three  chains  in  the  sum 
of  the  distances,  others  for  every  five  chains,  and  again  others  would 
require  it  to  be  within  one  link  for  every  ten  chains. 


138  A  MANUAL  OF  LAND  SURVEYING. 

As  a  check  against  errors  of  bearing,  a  back  sight 
should  be  taken  at  every  station,  and  the  reverse  bearing 
compared  with  the  corresponding  direct  bearing  of  that 
station.  If  the  two  are  found  to  differ  considerably,  both 
should  be  reviewed.  Let  us  now  see  how  small  an  error 
of  latitude  and  of  departure  we  have  in  the  present  ca.se. 

Sum  of  northings  =  10.74  +   5.05  +  10.23  =  26.02. 
"     "  southings  =   7.97  +  12.85+   5.34  =  26.16. 

Difference  of  latitudes  =  00.14  =  error  of  latitude. 
Sum  of  eastings  =  5.35  +   8.40  +  17.91  =  31.66. 
"     «  westings  =  8.99  +  11.99  +  10.59  =  31.57. 
Difference  of  departures  =  00.09  =  error  of  departure. 

The  above  errors  may  be  considered  reasonably  small 
for  a  field  of  the  size  of  the  present  one. 

In  practice,  some  of  the  courses  may  have  been  measured  over 
rough  or  uneven  ground,  and,  accordingly,  such  courses  should  beai 
a  larger  proportion  of  the  error. 

Some  of  the  bearings  may  have  been  taken  with  an  indistinct 
sight,  which  would  dictate  the  allotment  of  more  than  a  proportionate 
amount  of  the  error  to  them. 

Distances  as  measured  over  uneven  ground  are  liable  to  be  too  long. 
In  such  cases,  the  length  of  a  course  may  be  diminished  when  such 
change  would  favor  the  balancing.  Similarly,  a  doubtful  bearing  may 
be  changed,  if  the  error  should  appear  to  be  attributable  to  it. 

It  is  a  common  mistake  to  reverse  the  position  of  the  latitude  and 
departure  in  the  columns.  If  the  bearing  is  greater  than  45°  the 
departure  is  greater  than  the  latitude,  and  it  is  less  when  the  bearing 
is  less  than  45°.  Scan  the  columns  for  such  errors. 

11.  Balancing.— The  next  work  is  to  distribute  the 
errors  among  the  several  courses  in  proportion  to  their 
lengths,  in  accordance  with  the  following 

PRINCIPLE. — As  the  sum  of  the  lengths  of  all  the  courses 
is  to  the  length  of  each  course,  so  is  the  total  error  to  the 
error  of  that  course. 

This  operation  is  called  Balancing. 

Applying  the  above  principle,  we  divide  the  errors  by 
the  sum  of  the  lengths  of  all  the  courses  and  multiply  the 
quotients  by  the  length  of  each  course,  indicating  the 
products  as  positive  or  negative,  accordingly  as  they  are 
to  be  added  or  subtracted  in  making  the  required  correc- 
tion. 


COMPUTING   AREAS. 


139 


Thus,  00.14--84.92=00.00165;  and  00.09--84.92=OO.OOT06; 
00.00165X12=00.0198  or  +00.02;  and  00.00106X12=00.01272 
or  — 00.01,  to  the  nearest  link. 

In  the  same  manner,  by  multiplying  the  above  quotients 
by  the  lengths  of  the  other  courses,  the  correction  for 
them  is  readily  obtained. 

Collecting  results  thus  found,  we  have  the  following 

TABLE  I. 


Sta. 

Latitude. 

Departure. 

Cor.L 

CorD 

Balanced. 

N. 

s. 

E. 

W. 

N. 

S. 

E. 

W. 

1 

10.74 

5.35 

+.02 

—.01 

10.76 

5.34 

2 

5.05 

8.40 

+.02 

—.01 

5.07 

8.39 

3 

7.97 

17.91 

—.03 

—.02 

7.94 

17.89 

4 

12.85 

8.99 

—.03 

+.02 

12.82 

9.01 

5 

5.34 

11.99 

—.02 

+.01 

5.32 

12.00 

6 

10.23 

10.59 

+.02 

+.02 

10.25 

10.61 

We  next  find  the  Meridian  Distance  of  the  several 
stations. 

M.  D.  of  Sta.  l=Dep.  of  Course  1=5.34. 

M.  D  of  Sta.  2=M.  D.  of  Sta.  1+Dep.  of  C.  2 

=5.34+8.39=13.73. 
M.  D.  of  Sta.  3=M.  D.  of  Sta.  2+Dep.  of  C.  3 

=13.73+17.89=31.62. 
M.  D.  of  Sta.  4=M.  D.  of  Sta.  3—  Dep.  of  C.  4 

=31.62—9.01=22.61. 
M.  D.  of  Sta.  5=M.  D.  of  Sta.  4—  Dep.  of  C.  5 

=22.61—12.00=10.61. 
M.  D.  of  Sta.  0=M.  D.  of  Sta,  5—  Dep.  of  C.  6 

=10.61—10.61=0.00 


M.  D.  Of  C.  1=  M.D.Sta.O+M.D.Sta.l   =       0+5.34      = 


2= 
3= 


31.  D.  Sta.  I+M.  D.  Sta.  2 


M.  D.  Sta.  2+M.  D.  Sta. 


13.73+31.62 


=  9.535. 
.675. 


140 


A    MANUAL    OF    LAND    SURVEYING. 


M.  D. 
« 

M.  D.  Sta.  3+M.  D.  Sta.  4 

u     r 

a 

M.  D.  Sta.  4+M.  D.  Sta.  5 

"    6 

2 
M.  D.  Sta.  5+M.  D.  Sta.  0 

2 

31.62+22.61 


_9-  ~  .  g 


We  may  now  put  the  whole  matter  in  compact  tabu- 
lar form  as  follows. 


i 

S. 

Ml 

u 

S3 

1y      <V 
Q>     -3 

1  1 

CO 

"o 

PS 

i 

I 

cj 

OS 

CC 

JH     -w 

a*"      OJ 

p 

go 

e 

02 

M 

P 

O  i-5 

p 

& 

K 

03 

N. 

S. 

E. 

W. 

0 

N.  26i°E. 

12.00 

10.76 

5  34 

2.67 

28.7292 

1 

N.59°  E. 

9.80 

5.07 

8.39 

13,73 

9.535 

48.34245 

2 

S.66°  E. 

19.60 

7.94 

17.89 

31.62 

22.675 

180.0395 

a 

S.  35  °  W. 

15.68 

12.82 

9.01 

22.61 

27.115 

347.6143 

4 

S.  66  °  W. 

13.12 

5.32 

12.00 

10  61 

16.61 

88.3652 

5 

N.46°W. 

14.72 

10.25 

10.61 

0.00 

5.305 

54.37625 

26.0826.0831.62  31.62 


131-4479    616  0190 
131.4479 


484.5711 
=  Acres    48.45711 

In  this  example  the  area  of  the  tract  is  evidently 
equal  to  the  sum  of  the  areas  of  the  trapezoids  c  d  and 
e  based  on  courses  3,  4,  and  5  minus  the  sum  of  the 
areas  of  the  triangles  and  trapezoid  a  b  and  /  based  on 
courses  1,  2,  and  6. 

The  area  of  the  triangle  a  equals  the  M.  D.  of  course 
or  line  1  multiplied  by  its  latitude  =  2.67  x  10.76. 

The  area  of  the  trapezoid  b  equals  the  M.  D.  of  course 
2  multiplied  by  its  latitude  =  9.535  X  5.07. 

In  a  similar  manner  we  find  the  area  of  each  triangle 
and  trapezoid. 

Examples  for  Solution: 

The  f  Rowing  exainples  are  taken  from  the  field 
notes  of  the  original  United  States  Surveys  in  Michi- 
gan and  are  fair  samples  of  the  average  work  done  on 
the  government  >and  surveys.  The  meanders  of  lakes 
and  streams  are  run  for  the  purpose  of  finding  how 
much  dry  or  uncovered  land  is  contained  in  the  ad- 
jacent tract  to  be  paid  for  by  the  purchaser. 


COMPUTING    AREAS.  141 

Ex.  1.  Meanders  of  a  Lake  in  Section  5. 

Began  at  post  corner  to  Sections  4.  5,  8,  and  9,  thence 
in  Section  5,  N.  60°  W.  6.50  cli.  to  S.  K  Margin  of  Lake, 
thence  in  Sec.  5,  N.  25°  E.  4.00  ch.,  thence,  N.  51°  W.  5.0Q 
ch..  thence  N.  18°  W.  7.00  ch.,  thence  X.  3°  W.  7.00  ch.r 
thence  N.  63°  W.  10.00  ch.,  thence  S.  79°  W.  6.00  ch., 
thence  S.  7°  W.  13.00  ch.,  thence  S.  20°  E.  6.00  ch.,  thence 
S.  6°  W.  5.00  ch.,  thence  N.  78°  E.  14.00  ch.,  thence  S. 
27°  E.  5.00  ch.,  thence  N.  71°  E.  3.87  ch.  to  place  of  be- 
ginning on  margin  of  Lake. 

Find  the  area  of  the  lake.  Also  find  the  areas  of  the 
North  and  South  halves  respectively  of  the  quarter  sec- 
tion in  which  the  lake  lies,  on  the  supposition  that  the 
quarter  section  is  just  40  chains  square  and  that  the 
lines  are  run  with  the  same  variation  of  the  needle  as 
was  used  in  meandering  the  lake.  These  areas  are 
given  in  the  official  plat  as  follows:  North  i,  A.  66.18. 
South  i,  A.  55.92. 

2.  Find  the  area  of  the  lake  described  in  the  exam- 
ple 13,  page  109,  also  the  area  of  each  of  the  quarter- 
quarter  sections  adjoining  the  lake  in  the  south  half 
jf  Sections  11  and  12.    These  areas  are  marked  in  the 
official  plat  as  follows  :  In  Section  11,  S.  E.  i  of  S.  E.  i 
A.  31.50,  N.  E.  i  of  S.  E.  ±  A.  20.40.    In  Section  12.  S.  W. 
i  of  S.  W.  i  A.  37.61.  X.  W.  i  of  S.  W.  i  A.  27.10.     The 
meander  post  at  the  beginning  of  the  survey  is  14.00 
chains  North  from  the  Section  Corner. 

3.  Meander  of  a  Lake  in  section  2. 

Began  at  quarter  post  in  line  of  Sections  -2  and  li, 
thence  North  10.00  ch.,  to  S.  margin  of  Lake,  thence  in 
Sec.  2,  thence  S.  57°  E.  13.00  ch.,  thence  E.  3.00  ch., 
thence  N.  45°  E.  5.00  ch.,  thence  N.  4°  W.  6.00  ch.,  thence 
N.  70°  W.  15.00  ch.,  thence  S.  80°  W.  6.00  ch.,  thence 
S.  24i  E.  7.17  ch.,  to  place  of  beginning  in  margin  of 
Lake. 

Find  the  area  of  the  Lake  also  the  area  of  the  W.  i  of 
Ss.  E.  i'  of  Section  2  arid  of  the  S.  E.  i  of  the  S.  W.  i  ol 
the  Section.  The  first  is  given  on  the  official  plat  as  A. 
62.8S  and  the  latter  as  A.  38.95. 

• 


142  A   MAiOJAL  OF  LAKD  SUKVEYING. 

13.  Problem. —  Given  the  bearings  of  the  sides  of  a 
field,  to  find  the  bearings  when  the  field  is  supposed  to  be 
revolved  so  as  to  cause  one  of  the  sides  to  coincide  wiih  a 
meridian. 

EXAMPLES. 

1.  The  bearings  of  the  sides  of  a  field  are,  1st,  K 12°  E., 
2d,  N.  83£°  E.,  3d,  S.  21°  W.,  and  4th,  N.  47°  W.   What  will 
the  bearings  be,  if  the  field  be  supposed  to  be  revolved  so 
as  to  cause  the  first  side  to  be  on  a  meridian  ? 

Ans.~  1st,  N.,  2d,  N.  71£°  E.,  3d,  S.  9°  W.,  and  4th, 
N.  59°  W. 

SUGGESTION.— Suppose  the  field  to  be  revolved  toward  the  left, 
through  an  angle  of  12°.  Accordingly,  each  bearing  would  be  changed 
by  that  amount.  The  readings  of  the  new  bearings  are  readily  deter- 
mined by  inspection. 

2.  The  bearings  of  the  sides  of  a  field  are  1st  S.  3|°  W., 
2d  N.  86|°  W.,  3d  N.  16£°  E.,  and  4th  E.    Required  the 
new  bearings  when  the  first  side  is  made  to  coincide  with 
the  meridian. 

Ans.— 1st  S.,  2d  W.,  3d  X.  13°  E.,  and  4th  N.  86|°  E. 

3.  The  bearings  of  the  sides  of  a  field  are  1st  S.  20°  W., 
2d  S.  70°  W.,  3d  N.  31°  W.,  4th  N.  45°  E.,  and  5th  S.  60°  E. 
Required  the  new  bearings  when  the  third  side  is  made 
to  coincide  with  the  meridian. 

Ans.— 1st  S.  51°  W.,  2d  N.  79°  W.,  3d  X.,  4th  N.  76°  E., 
and  5th  S.  29°  E. 

4.  The  bearings  of  the  sides  of  a  field  are,  1st  K  45°  E., 
2d  IS.  30°  W.,  3d  S.  5°  E.,  4th  W.,  and  5th  N.  20°  E.    What 
will  the  bearings  become,  if  the  field  be  revolved  so  as  to 
bring  the  third  side  to  the  meridian  ? 

Ans.— 1st  N.  50°  E.,  2d  S.  35°  W.,  3d  S.,  4th  N.  85°  W., 
5th  N.  25°  E. 

5.  The  bearings  of  the  sides  of  a  field  are,  1st  E.,  2d 
N.  9°  E.,  3d  S.  69°  E.,  4th  S.  66°  E.,  5th  S.  42°  W.,  6tb 
S.  75°  W.,  7th  N.  39°  W.,  and  8th  N.  42°  E.    What  will  the 
bearings  become,  if  the  field  be  revolved  so  as  to  cause 
the  fourth  side  to  coincide  with  the  meridian  ? 

Ans.— 1st  S.  24°  E.,  2d  N.  75°  E.,  3d  IS.  3°  E.,  4th  S.,  5th 
K.  72°  W.,  etc. 

Additional  exercises  may  be  formed  from  the  above  by  requiring 
different  sides  to  be  brought  to  coincide  with  the  meridian. 


COMPUTING  ABEAS. 


143 


RULE.— Change  each  bearing  agreeably  with  the  direc- 
tion in  which  the  field  is  mpposed  to  be  r:  wived  by  an 
amount  equal  to  the  bearing  of  the  side  which  is  brought 
to  the  meridian,  and  express  the  result  in  accordance  with 
the  proper  form  of  denoting  bearings. 

6.  What  were  the  bearings  of  the  sides  of  a  field  which 
are  now  N.  16|°  E.,  E.,  S.  3£°  W.,  and  X.  86|°  W.,  the  vari- 
ation of  the  needle  having  changed  2£°  toward  the  west 
since  the  former  survey  ? 

Supplying  Omissions.  — From  inaccessibility  of 
lines  and  sometimes  from  accident,  omissions  may  occur 
in  the  field  notes  of  a  survey.  In  a  closed  survey,  any 
two  omissions  may,  in  general,  be  supplied  by  computa- 
tion. It  is,  however,  desirable  to  avoid  as  far  as  possible 
the  necessity  of  supplying  omissions  in  this  manner,  since 
it  infringes  upon  the  tests  which  otherwise  serve  to  verify 
the  work. 

The  several  cases  which  may  occur  are  presented  in  the 
following  problems: 

14.   Prob.  1.  To  find  an  omitted  bearing  and  distance. 
CASE  1. —  When  the  omissions  pertain  to  the  same  course. 

In  a  closed  survey,  the  sum  of  the  northings  should 
equal  the  sum  of  the  southings;  and  the  sum  of  the  east- 
ings should  equal  the  sum  of  the  westings.  The  defect 
of  these  equalities  in  the  present  case  must  be  on  the  one 
hand  the  latitude  and  on  the  other  the  departure  of  the 
omitted  course. 


Example. — 


Sta. 

Bearing.' 

Dist. 

Lat. 

Dep. 

A 

N.31°W. 

9,40 

-B.057 

—  4.841 

B 

N.45°E. 

9.30 

-f-6.576 

-f  6.576 

C 

Oniii 

ted. 

E 

S.  20°  W. 

5.30 

—4.980 

—  1.813 

F 

S.  70°  W. 

10.90 

-3.728 

—10.243 

144 


A  MANUAL  OF  LAND  SURVEYING. 


Solution.  —  Sum  of 

northings  =  14.633 
of  southings  =   8.708 


Diff.  =  CG  =   5.925 

Sum  of 

westings  =  16.897 
of  eastings  =   6.576 


Diff.  =  GE  =  10.321 


FIG.  48. 


The  latitude  of  the  omitted  course  is  thus  a  southing 
and  its  departure,  an  easting.  Its  bearing  is  therefore 
S.  — °  E. 

To  find  the  bearing  or  angle  GCE,  we  have 

GE      10.321 

tan  GCE  =  • —  = =  1.74194. 

CG       5.925 

Whence,  GCE  *==  60°  8';  or  the  required  bearing  ii 
«.  60°  V  E. 

To  find  the  distance  CE,  we  have 

CE  =  (5.9252  +  10.3212)*  =  12.00. 

REMARK.— It  will  be  noticed  that  a  plat  of  the  field  may  be  made, 
and  the  area  found  without  supplying  the  omissions. 

CASE  2. — When  the  omissions  pertain  to  different 
courses. 

If  the  field  be  supposed  to  be  revolved  until  the  side 
whose  length  is  omitted  becomes  a  meridian,  the  given 
bearings  being  changed  accordingly  (Art.  13,  Prob.),  then, 
since  the  departure  of  the  side  made  a  meridian  is  0,  the 
difference  between  the  sums  of  the  eastings  and  westings 
of  the  other  courses  is  the  departure,  in  its  new  position, 
of  the  side  whose  bearing  is  omitted. 


COMPUTING  AREAS. 


145 


Knowing  the  length  and  th£  departure  of  this  side,  its 
latitude  and  bearing  may  be  found,  (Art.  8). 

The  difference  between  the  sums  of  the  northings  and 
southings  of  the  courses  in  their  new  positions,  is  the 
length  of  the  side  which  was  made  a  meridian. 

Example. — 


Sta. 

Bearing,  g-^ 

Distance. 

Lat. 

Dep. 

A 

N.  20°  E. 

North. 

Omitted. 

0.0000 

B 

X.45°E. 

N.  25°  E. 

8.00 

+7.2505 

+3.3809 

C 

S.  "30°  W. 

S.  10°W. 

5.00 

—4.9240 

—0.8682 

D 

Omitted. 

7.20 

E 

West. 

S.  70°  W. 

5.92 

—2.0248 

—5.5630 

Solution.— Sum  of  eastings  ==  3.3809 
"     "  westings  =  6.4312 


Difference  =  3.0503  (an  easting). 

Latitude  of  DE  =  (7.202  —  3.05032)*  =  6.5219  (a  southing). 
Sine  of  changed  bearing  of  DE  =  3.0503  -*-  7.20  =  0.42365. 
Whence  "  "  DE  is   S.  25°  V  E. 

Whence  original    "        "  DE  was  S.   5°  V  E. 
Sum  of  northings  =    7.2505 
"     "    southings  *-=  13.4707 

Difference  =   6.22     =  length  of  AB. 

REMARK.— It  is  sometimes  doubtful  whether  the  latitude  of  the 
course  whose  beariug  is  omitted  is  a  nortliing  or  a  southing. 

In  the  present  case,  the  question  is  determined  by  a  simple  inspec- 
tion of  the  latitudes,  since  the  sum  of  the  southings  is  less  than  the 
sum  of  the  northings,  without  considering  the  northing  of  the  first 
course. 

In  other  cases,  there  may  be  two  sets  of  values  of  the  omitted  parts. 
with  either  of  which  the  problem  is  satisfied. 

Practical^,  however,  the  ambiguity  is  removed  by  a  general  knowl- 
edge wlueh  the  surveyor  has  of  the  directions  of  the  lines. 

10 


146 


A  MANUAL  OF  LAND  SURVEYING. 


15.  Prob.  2.  To  find  the  omitted  lengths  of  two  courses. 
CASE  1. —  When  the  courses  are  consecutive. 

The  bearing  and  length  of  a  line  which  would  close  a 
survey,  leaving  out  the  unknown  sides,  may  be  found  by 
Prob.  1,  Case  1.  This  line  and  the  unknown  sides  form  a 
triangle  in  which  the  angles,  as  found  from  the  given 
bearings,  and  the  length  of  one  side  are  known.  The 
lengths  of  the  other  sides  may  therefore  be  computed. 

The  procedure  will  be  readily  worked  out  by  the  stu- 
dent, without  illustration. 

CASE  2. —  When  the  courses  are  not  consecutive. 

This  case  may  be  treated  in  the  same  manner  as  the 
preceding. 

Or,  we  may  suppose  the  field  to  be  revolved  so  as  to 
make  one  of  the  sides  whose  length  is  omitted,  a  merid- 
ian, the  bearings  of  the  other  sides  being  changed  accord- 
ingly. 

We  may  then  find  the  difference  of  the  sums  of  the 
eastings  and  westings,  which  will  be  the  departure,  in  its 
new  position,  of  the  other  side  whose  length  is  wanting. 

Having  the  bearing  of  that  side  and  its  departure,  its 
length  and  latitude  may  be  found.  Finding  the  differ- 
ence between'  the  sums  of  the  northings  and  southings, 
we  obtain  the  length  of  the  side  which  was  made  a 
meridian. 

Example. — 


Sta. 

Bearing. 

Changed 
Bearings. 

Distance. 

Lat. 

Dep. 

A 

N.  15°  E. 

Nv30°  W. 

5.00 

-f  4.33 

—2.50 

B 

N.  45°  E. 

North.     ' 

Omitted. 

0.00 

C 

S.  55°  E. 

N.  80°  E, 

10.05 

4-  1-75 

+0.90 

D 

S.  15°  W. 

S.  30°  E. 

12.25 

;—  10.61 

+6.12 

E 

S.75°W. 

S.  30°  W. 

Omitted. 

F 

N.33%°W. 

N.785£°W. 

9.96 

-f  1.95 

—9.77 

COMPUTING   AREAS.    N  147 

Sum  of  eastings  =  16.02 
"      "  westings  =  12.27 

Difference  =   3.75  =  Dist.  X  sin  30°. 
Whence,  length  of  EF  =  3.75  -f-  0.5  =  7.50. 
Lat.  EF  =  3.75  +  tan  30°  *=  6.50. 
Sum  of  northings  =   8.03 
"      "   southings  =  17.11 

Difference  =   9.08  =  length  of  BC. 

REMARK.— If  the  sides  whose  lengths  are  omitted  are  parallel,  the 
problem  is  indeterminate. 

16.  Prob.  3.  To  find  the  omitted  bearings  of  two 
courses. 

We  find,  (Prob.  1,  Case  1)  the  bearing  and  length  of  a 
line  which  would  close  a  survey,  having  the  lines  whose 
bearings  are  given  as  the  other  sides. 

The  line  thus  found  and  the  two  lines  whose  bearings 
are  omitted  form  a  triangle.  The  lengths  of  the  sides  of 
the  triangle  being  known,  its  angles  maybe  found;  and 
from  the  angles  and  the  bearing  of  one  of  the  sides  the 
bearings  of  the  other  sides  may  be  found. 

The  closing  line  and  the  triangle  are  illustrated  by  the 
diagram  accompanying  the  following 

Example.— 


Sta. 

Bearing. 

Dist. 

Lat. 

pep. 

A 

N.  15°  E. 

5.00 

+  4.8296 

+1.2941 

B 

Omitted. 

9.08 

c 

S.  55°  E. 

10.05 

—  5.7645 

+8.2325 

D 

S.  15°  W. 

12.25 

—11.8327 

—3.1705 

E 

Omitted. 

7.50 

F 

N.  33?i°  \V. 

9.96 

-f  8.2814 

—5.5334 

I 

1(1 


148 


A   MANUAL  OF  LAND  SURVEYING. 


The  side  EF,  without  change  of  bearing,  is  represented 

by  CG.     BG  is  the 
closing   line   of    the 
field    ABGHF,    in 
& , — "      /          ^X^    ,  which  we  have 

Sum  of 

northings  =  13.1110 
southings  =  17.5972 

Difference  =  4.4862 
(a  northing). 


Sum  of 

eastings 
westings 


9.5266 
8.7039 


Difference  =  0.8227 
(a  westing.) 


Whence  (Prob.  1),  bearing  BG  is  K.  10°  23'  30"  W.,  and 
length  BG  is  4.56. 

In  the  triangle  BGC,  BC  =  9.08  and  CG  =  EF '=  7.50. 

Solving  the  triangle,  we  find 

angle  GBC  =  55°  25'  40",  ana  angle  BGC  =  94°  31'  49". 

Whence,  bearing  BC  is  N.  45°  2'  10"  E.,  and  bearing 
EF  is  S.  75°  V  41"  W. 

REMARK.— The  problem  may  possibly  have  two  solutions,  accord- 
ingly as  the  triangle  may  fall  on  either  side  of  the  closing  line.  The 
ambiguity  is,  however,  practically  unimportant. 

Exercises. — To  be  made  by  the  student  in  the  field. 

17.  Most  of  the  foregoing  problems  for  finding  areas 
may  be  simplified  and  much  labor  saved  in  calculation, 
by  reducing  the  irregular  polygons  and  oblique  triangles 
to  right  triangles  and  trapezoids  on  the  plat,  and  taking 
their  dimensions  by  direct  measurements  from  the  plat, 
instead  of  calculating  them.  If  the  plat  is  made  on  a 
large  enough  scale — showing  not  more  than  four  chains 


COMPUTING  AREAS. 


149 


to  the  inch — and  the  drafting  is  carefully  done,  tne  meas- 
ures on  the  plat  will  be  very  nearly  if  not  quite  as  good 
as  those  taken  on  the  ground,  and  will  give  results  suffi- 
ciently close  for  most  purposes. 

IST  METHOD. — Draw  a  diagonal  between  two  distant 
angles  of  the  figure,  and  perpendiculars  to  it  from  the 
other  angles. 


FIG.  so. 


2ND  METHOD. — Reduce  the  figure  to  a  single  equivalent 
triangle. 


FIG.  51. 


150 


A  MANUAL  OF  LAND  SUEVEYING. 


1.  To  reduce  the  trapezium  abed  (Fig.  51)  to  its  equiva- 
lent triangle. 

Produce  the  line  ab  an  indefinite  distance.  With  the 
parallel  ruler,  or  straight  edge  and  triangle,  find  the  point 
e,  where  a  line  through  d  parallel  to  ca  intersects  the  line 
ab.  Draw  the  line  ec,  intersecting  ad  at  g. 

Then  the  triangle  ecb  is  equivalent  to  the  trapezium 
abed,  for  the  triangles  acd  and  ace,  having  the  same  base 
ac  and  equal  altitudes,  are  equal;  and  the  triangle  aeg 
being  taken  from  both  leaves  the  triangle  eag,  which  is 
added  to  the  original  figure,  equal  to  the  triangle  cdg, 
which  is  taken  from  it. 

The  perpendicular  may  now  be  drawn  from  c,  and  the 
base  eb  and  altitude  fc  measured  on  the  plat. 

2.  By  an  extension  of  the  same  process,  any  polygon 
may  be  reduced  to  one  or  more  equivalent  triangles.    It 
will  frequently  be  found  convenient  to  divide  the  figure 
into  two  or  more  parts,  and  reduce  the  sides  separately. 
The  process  is  indicated  in  Figure  52. 


FIG.  52. 


Let  abcdefgh  be  the  polygon  to  be  reduced.     Extend 
one  side,  as  ab,  indefinitely  for  a  base.    From  c  draw  ci 


COMPCTTIXG   AREAS.  151 

parallel  to  bd.  From  d  draw  dk  parallel  to  ei.  From  e 
draw  el  parallel  to  fk.  Having  selected  /  as  the  vertex 
of  the  triangle,  we  next  draw/?  for  one  of  its  sides. 

Next,  from  h  draw  Jim  parallel  to  ga. 

From  g  draw  gn  parallel  to  fin. 

From/ draw  fn  for  the  third  side  of  the  triangle,  and 
fo,  its  altitude. 

The  triangle  fin  is  equivalent  to  the  polygon  abcdefgh. 
It  is  best  to  draw  all  these  lines  lightly  on  the  plat,  to 
avoid  errors. 

If  we  consider  each  point,  i,  k,  I,  marked  in  succession 
on  the  base  as  an  angle  of  the  polygon,  which  it  is  until 
its  successor  is  located,  we  have  the  following 

GENERAL  RULE. — Extend  one  side  indefinitely  as  a 
base.  Commencing  at  the  first  angle  from  the  base,  draw 
from  it  to  the  base  a  line  parallel  to  a  line  joining  the  two 
adjacent  angles  of  the  polygon.  Continue  draioing  lines 
to  the  base  from  each  angle  in  succession  as  far  as  re- 
quired. Join  the  last  angle  from  which  a  parallel  was 
taken,  with  the  last  point  of  intersection  on  the  base,  for 
a  side  of  the  final  triangle. 

It  is  sometimes  more  convenient  not  to  produce  one  of 
the  lines  of  the  figure  for  a  base,  but  to  draw  a  perpen- 
dicular to  it  from  one  end  or  from  the  end  produced. 
The  same  rule  applies. 

18.  The  preceding  methods  of  taking  measurements 
from  the  plat  are  found  very  convenient  in  estimating 
the  area  of  land  benefited  by  drainage,  under  the  drain 
laws.  Surveyors  are  frequently  called  on  to  make  surveys 
and  maps  of  drainage  districts,  showing  the  location  of 
the  drains  and  the  location  and  area  of  the  lands,  belong- 
ing to  the  various  owners,  which  will  be  benefited  by  the 
drainage.  In  most,  if  not  all  these  cases,  no  man  can  tell, 
either  before  or  after  the  drainage  has  been  executed,  just 
exactly  where  the  dividing  line  is,  between  land  which  is 
benefited  and  that  which  is  not  benefited.  For  this  rea- 
son a  rapid  survey  of  the  approximate  line,  by  stadia 


152  A   MANUAL   OF   LAND   SURVEYING. 

measures,  is  just  as  good  as  the  most  elaborate  work  with 
the  chain  or  tape.  The  one  is  likely  to. get  as  near  the 
true  dividing  line  as  the  other. 

The  writer  has  found  the  following  method  to  work 
well  in  his  practice.  Suppose  a  tract  of  marsh  or  swamp 
is  to  be  measured  and  mapped,  having  more  or  less 
cleared  upland  around  it: 

Assume  some  line  as  a  base.  A  section  line  or  quarter 
line  of  the  United  States  Survey  answers  well  for  this 
purpose.  From  this  base  run  a  broken  line  around  the 
swamp  wherever  it  is  most  convenient  to  do  so.  Set  a 
stake  at  each  angle  in  the  line.  Note  the  length  of  each 
course  and  the  angle  which  it  makes  with  the  common 
base,  as  described  on  pagelOl . 

When  the  circuit  of  the  swamp  has  been  made,  and  the 
transit  again  set  up  at  the  starting  point,  the  work  will 
prove  itself.  After  taking  a  back  sight  on  the  last  sta- 
tion and  pointing  the  telescope  along  the  base  line,  if  the 
work  has  all  been  correctly  done,  the  vernier  should  give 
the  same  reading  as  it  did  to  start  with,  showing  that 
just  360°  have  been  passed  around. 

In  passing  around  the  swamp  an  assistant  with  the 
stadia  rod  follows  its  margin,  setting  up  his  rod  at  every 
point  where  it  changes  its  general  direction.  The  transit- 
man  notes  down  the  direction  of  each  point  at  which 
the  rod  is  set  up,  by  its  angle  from  the  base  line  and  its 
distance  from  tl^e  transit  as  read  off  from  the  rod. 

When  as  many  points  are  iaken  as  are  convenient  from 
one  station,  the  transit  is  moved  up  to  the  next  one,  and 
the  operation  continued  till  all  the  desirable  points  are 
located.  This  being  done  in  the  field,  they  are  reproduced 
on  the  plat  on  a  scale  large  enough  to  permit  measure- 
ments on  the  plat  with  a  reasonable  degree  of  accuracy. 
The  points  along  the  margin  of  the  swamp  having  been 
laid  down  on  the  plat,  are  connected  by  straight  lines,  and 
all  intersecting  farm  lines  or  other  points  of  interest  are 
also  laid  down. 


COMPUTING    AREAS.  153 

We  now  have  a  map,  showing  as  correctly  as  it  is  pos- 
sible to  do  so,  the  location  of  the  swamp  on  each  man's 
land.  The  areas  of  the  several  tracts  are  found  by  taking 
the  parallel  rule  and  needle  point  and  reducing  these 
irregular  polygons  to  their  equivalent  triangles  and  rect- 
angles, making  the  necessary  measures  on  the  plat  and 
computing  the  areas  from  these  measures. 

19.  Division  and  Partition  of  Land.—  The  sur- 

veyor is  sometimes  called  on  to  divide  areas  into  portions 
having  a  specified  relation  to  each  other,  or  to  part  off 
from  a  field  a  given  number  of  acres  by  a  line  fulfilling 
some  specified  condition  with  respect  to  the  field  divided. 

/•          «w         %£?» 

There  is  a  great  variety  of  these  problems,  most  of 
which  occur  very  rarely  in  the  surveyor's  practice.  A 
few  of  those  which  occur  most  frequently  are  given. 

Prob.  1.  —  To  divide  a  triangle  into  parts  having  a 
given  ratio. 

CASE  \—By  lines  from  an  angle. 

Solution.—  Let  ABC  be  any  triangle,  and  suppose  it  is 
required  to  divide  it  by  a  line  from 
B,  into  two  parts  having  the  ratio 
of  m  to  n. 

Let  BD  be  the  line  of  division,  so 
c  that  ABD  :  DBG  ::  m  :  n  (1) 
But  ABD  :  DBG  ::  AD  :  DO  (2) 

Combining  (1)  and  (2),  we  have 

AD  :  DC  ::  m  :  n, 
whence,  AD  :  AC  ::  m  :  m+n, 


whence,  AD  ^=  -  .    Similarly,  DC  = 

wi  +  n  m  +  n 

Measure  the  distance  AD  thus  found,  and  run  the  line 
BD. 


154  A  MANUAL  OF  LAND  SURVEYING. 

If  the  triangle  were  to  be  divided  into  three  parts  in 
the  ratio  of  m  :  n  :  p,  we  should  have 

mXAC                          nXAC 
AD  = and  DE  =  -       . 

m  +  n  -\-  p  m  4-  n  +  p 

Cor.— To  part  off  by  a  line,  as  BD,  a  given  area  a,  we 

a  X  AC 

have  AD  :  AC  : :  a  :  area  ABC,  whence  AD  — . 

area  ABC 

Examples.— I.  Find  the  measurements  required  to  di- 
vide a  trianglar  field  by  lines  from  an  angle  to  a  side 
whose  length  is  12.30 -ch.,  into  parts  to  each  other  as  2,  3 
and  4. 

2.  Find  the  measurement  required  to  part  off  3.5  acres 
from  a  triangular  field  a  side  of  which  is  18.50  ch.,  and 
a  perpendicular  thereupon  from  the  opposite  angle  is 
10.40  ch. 

CASE  2.~ By  lines  parallel  to  a  side. 

Solution.  —  Let  D  be  the  point  in  the  side  AB  from 
which  a  line  parallel  to  BC  shall 
divide  ABC  so  that  ADE  :  DECS 
::  m  :n.     Then 
/-  — Sj  ADE  :  ABC  : :  m  :  m  +  n. 

<&—          <"•    But  ADE  :  ABC  ::  AD2  :  AB*, 

FIG.  54.  whence,  AD2  :  AB2  : :  m  :  m  +  n, 

(      m 

giving  AD  =  AB  ]  

(m  -f  n 

Measure  the  distance  AD  thus  found,  and  run  DE 
parallel  to  BC. 

If  the  triangle  is  required  to  be  divided  into  three  parts 
in  the  ratio  of  m  :  n  :  p,  we  should  have 

m         )  x  (     m  -\-  n 
f    tm&AF  =  AB 


m-\-n-\-p 
Cor.  1.— To  part  off  a  triangle,  as  ADE,  of  given  area  a 


rOMFUTING   AREAS. 
we  have  AD  =  AE 


— r 

^ .       j 


Cor.  2.—  To  part  off  a  quadrilateral,  as  DECB,  of  given 
area,  a',  we  may  find  by  Cor.  1  the  distance  AD  required 
to  part  off  a  triangle  of  the  area  ABC  —  •  a'  and  measure 
BD  =  BA  —  AD. 

Examples.  —  1.  Find  the  measurement  for  dividing  a 
triangular  field  of  12  A.  into  parts  in  the  ratio  of  4  to  5 
by  a  parallel  run  from  a  point  in  a  side  whose  length  is 
10.35  ch. 

2.  Find  measurements  for  dividing   by  parallels,  the 
above  field  into  three  equivalent  parts. 

3.  Find  measurement  for  parting  off  from  the  same 
field  by  a  parallel,  a  triangle  of  5  A.;  a  quadrilateral  of 


CASE  Z.—By  lines  perpendicular  to  a  side. 

Solution.  —  Let  ABC  be  a  triangle  required  to  be  divided 
by  a  perpendicular  to  AC,  into  parts 
having  the  ratio  of  ra  to  n. 

Let  EF  be  the  line  of  division,  so 
that  AEF  :  EBCF  :  :  m  :  n,  or 
AEF  :  ABC  :  :  m  :  m  -f  n.  (1) 
Let  BD  De  a  perpendicular  upon  A  C 
Then  AEF:ABC::AFXEF:ACXBD::m:m  +  n.  (2) 
From  similar  triangles,  AF  :  EF  :  :  AD  :  BD, 

AFX  3D 

whence,  EF  =  -  . 
AD 

Substituting  this  value  of  EF  in  (2),  we  have 


:  AC  X  BD  ::  m  :  m  +  w, 

AD  or  AF*  :ACXAD  ::m:m  +  n 


whence,  AF  = 


Find  AD  and  then  AF.    Measure  the  distance  AF  and 
run  FE  perpendicular  to  AC. 


156  A    MANUAL    OF   LAND    SURVEYING. 

Similarly,  may  be  found  the  distances  to  perpendiculars 
dividing  the  triangle  into  three  or  more  parts  having  a 
given  ratio. 

Oor.  —  To  part  ofE  a  triangle,  as  AEF,  of  given  area,  a, 
XAD  X 


we  have  AF  = 


area,  A  BCD 


a)* 

[•  . 
) 


The  distance  AF  to  a  perpendicular  which  shall  part 
off  a  triangle  AEF  =  a,  may  be  found  otherwise,  as 
follows:  triangle  AEF  =  %AF  X  EF  =  a,  and  EF  = 


AF  X  tan  A.    Whence,  AF  = 

'  tan  A 

Examples. — 1.  The  bearings  and  lengths  of  two  sides 
of  a  triangular  field  from  the  same  corner  are  N.  20°  E., 
15  ch.,  and  N  50°  E.,  20  ch.  Required  the  measurement 
from  that  corner  to  a  perpendicular  upon  the  longer  side 
which  shall  divide  the  field  into  two  parts  having  the  ra- 
tio of  2  to  3. 

2.  Required  the  measurement  to  a  perpendicular  which 
shall  divide  the  above  field  into  two  equivalent  parts; 
into  three  equivalent  parts. 

3.  Required  the  measurement  to  a  perpendicular  which 
shall  part  off  from  the  same  field  a  triangle  of  4  A.;  a 
quadrilateral  of  5  A. 

2O.  Prob.  2.  To  divide  a  trapezoid  into  parts  having 
a  given  ratio. 

CASE  I.— By  lines  dividing  the  bases  proportionally. 

Solution.— Let  ABCD  be  any  trapezoid  required  to  be 
divided  into  parts  having  the  ratio  Qfm\n:p. 
This  is  done  in  the  easiest  manner  by  dividing  each 
base  into  parts  having  the  ratio  to 
each  other  as  m,  n  and  p,  and  join- 
ing  the   corresponding   points   of 
division.    The  measurements  nec- 
essary to  find.the  points  of  division 
FIG.  56.  are: 


COMPUTING  AREAS.  157 

m  X  BC  nXBC 


m  +  n+p  m  +  n-\-p  m-\-n+p 

and  FH  = 


Cor.— To  part  off  a  given  area  a  by  a  line,  as  EF, 
which  shall  divide  the  bases  proportionately,  we  have 

aXBC  aXAD 

BE  = and  AF  = . 

area  ABCD  area  ABCD 

Examples.— -1.  Given  AD,  N.  80°  E.,  '12.60  ch.,  AB, 
N.  10|°  E.,  8.12  ch.,  and  BC,  X.  80°  E.,  10.34  ch.,  to  find  the 
measurements  required  in  dividing  the  field  into  parts 
having  the  ratio  of  4  to  7,  by  a  line  dividing  the  parallel 
sides  proportionally. 

2.  Find  the  measurements  for  parting  off  from  the 
above  field  an  area  of  5  A.,  by  a  line  dividing  the  parallel 
sides  proportionally. 

CASE  2. — By  lines  parallel  to  the  'bases. 

A  Solution. — Let  ABCD  be  a  trape- 
zoid  to  be  divided  into  parts  in  the 
ratio  of  m  to  n,  by  a  line  parallel  to 
v     ** 

1^  \         Suppose  EF  to  be  the  required 

^ line  of  division,  so  that 
1  :c-  57-  EBCF :  AEFD  ::m:n. 

Regarding  the  sides  AB  and  DC  as  prolonged  to  meet 
•at  O,  we  have  OAD  :  OBC  : :  AD2 :  BC2, 
whence,  OAD  —  OBC, 
or  ABCD:  OBC'.:  AD*  —  BC2:  BC2.  (1) 

Similarly,  we  have  EBCF:  OBC::  EF2—BCZ:  BC2.  (2) 
Combining  (1)  and  (2),  ABCD:  EBCF:: 


158  A  MANUAL  OF  LAND   SURVEYING. 


Cm  X 

whence  EF  =  \-  (a) 

I  m  -\-n  ) 

Supposing  BH  to  be  parallel  to  CD,  the  triangles  ABH 
and  EBG  give  AB  :  AH  ::  EB  :  EG, 

or    AB  :  AD  —  BC  ::  EB  :  EF  —  BC. 

AB(EF  —  BC} 

Whence,  EB  =  -  -  .    (6). 

AD  —  BC 

Thus,  first  finding  EB  by  formula  (a),  we  can  then  find 
EB  by  formula  (6),  and  measuring  that  distance  from  B, 
we  may  run  EF  parallel  to  BC,  dividing  the  trapezoid  as 
required. 

Similarly,  a  trapezoid  may  be  divided  in  three  or  more 
parts  having  a  given  ratio.  Indeed,  the  above  formulas 
may  be  directly  applied  to  that  purpose  by  making  a 
simple  substitution. 

Cor.  —  To  part  off  a  trapezoid  of  given  area  a,  adjoining 
BC,  we  obtain  from  formula  (a) 

(  a  X  AD2  -f  (area  ABCD  —  a)  BO2  )  K 
EF  =  \  - 

I  area  ABCD  ) 

'.The  distance  BE  is  then  found  from  formula  (6). 

Examples.  —  1.  Given  a  trapezoidal  field  ABCD  in  which 
AB  is  an  east  and  west  line,  9  ch.,  BC  a  north  and 
south  line.  5.19  ch.,  and  AD  a  north  and  south  line,  8  ch., 
it  is  required  to  run  a  north  and  south  line  dividing  the 
field  so  that  the  parts  on  BC  and  AD  shall  have  the  ratio 
of  2  to  3.  , 

2.  Find  the  measurement  from  A  to  part  off  from  the 
above  field  by  a  north  and  south  line  an  area  of  3  A.  ad- 
joining AD. 

CASE  3.  —  By  lines  perpendicular  to  the  bases 


COMPUTING  ABEAS.  159- 

Solution.  —  Let  ABCD  be  a  trapezoid  to  be  divided  into 
parts  in  the  ratio  of  m  to.  n  by  a  line 
perpendicular  to  AD. 

L\  Let  EF  be  the  line  joining  the 
middle  points  of  the  non-parallel 
sides  AB  and  CD.  We  divide  EF, 


FIG.  57.  as  a£  ^  into  two  parts  having  the 

ratio  of  m  to  n,  and  through  &  run  HI  perpendicular  to 
AD. 

To  find  the  point  G  on  the  ground,  we  have  the  forni- 

m(BC  +  AD) 
ula  EG  =  -  •  --  .    Whence,  measuring  from  E  the 

2  (m  -f  w) 

distance  .##  on  the  bearing  of  13  C,  we  have  the  point 
sought. 

Cor.—  To  part  off  a  given  area  a,  by  a  line  perpendic- 

a  (BC  -f  AD) 

ular  to  the  bases,  we  have  EG  =  -----  , 

2  XareaJ^OD 

Or,  denoting  the  altitude  of  the  trapezoid  by  A,  we 

a  a 

have  J57<?  =  —  =  -  . 


h       AB  X  sin  A 

The  point  /  or  H  may  be  found  by  the  formula 
AI=EG  4-  AE  X  cos  A,  or  BH  =  EG  —  EB  X  cos  4. 

Examples.—  1.  Given  J.D,  E.  20  ch.,  AB,  X.  15°  E., 
9.50  ch.,  and  BC,  E.  12  ch.,  required  the  measurement  for 
dividing  the  field  by  a  perpendicular  to  A  D  into  two  parts 
having  the  ratio  of  m  to  n. 

2.  Required  the  measurement  for  parting  off  from  the 
above  field,  by  a  perpendicular  to  AD,  an  area  of  4  A. 
adjoining  AB. 

21.    Prob.  3.  —  To    Jirtde    a    trapezium    into  parts 
a  given  ratio. 


CASE  1.  —  By  lines  from  an  angle. 


160  A   MANUAL   OF   LAND   SURVEYING. 

Solution. — Let  ABCD  be  a  trapezium  to  be  divided  into 
two  parts  having  the  ratio  of  m 
to  n,  by  a  line  from  C. 

We  draw  AC,  and  from  B  draw 
a  line  parallel  to  AC,  meeting  DA 
produced  at  E.    We  then  divide 
ED,  as  at  F,  into  the  parts  EF 
FlG- E9-  and  FD,  having  the  ratio  of  m  to 

n.    The  line  CF  divides  the  trapezium  as  required.    That 
is,  ABCF  :  FCD  ::  m  :  n,  or  ABCF  :  ABCD  : :  m  :  m  -f  n. 

SCH.— The  above  solution  is  readily  executed  on  the  ground. 

In  a  similar  manner  a  trapezium  may  be  divided  into 
any  number  of  parts  having  a  given  ratio. 

The  point  F  may  be  otherwise  found  as  follows: 

n  X  ABCD 
The  triangle  DC  F  = , 


and  again,  DCF  = 


m      n 

DC  X  sin  D  X  DF 


2 

2n  X  ABCD 

Whence,  DF  = . 

DC  (m  +  n)  sin  D 

Cor.— To  part  off  a  triangle,  as  DCF,  of  given  area  a, 
2a 

we  have  DF  -— . 

DC  X  sin  D 

Examples— 1.  Given  AB,  N.  8°  W.,  7.60  ch.,  BC 
N.  76i°  E.,  10.21  ch.,  CD,  S.,  11.40  ch.,  and  DA,  N.  80J°  W. 
9.00  ch.  Required  the  measurement  lor  locating  a  line 
CF  which  shall  divide  ABCD  into  the  parts  ABCF  and 
FCD,  to  each  other,  respectively,  as  2  to  3. 

2.  Required  the  measurement  for  parting  off  from  the 
above  field  a  triangle  DCF  of  1 . 6  A. 


COMPUTING   AREAS.  101 

CASE  %.—By  lines  parallel  to  a  side. 

Solution.— Let  it  be  required  to  divide  a  trapezium,  as 
ABCD,  by  a  line,  as  EF,  parallel 
to  AD,  into  two  parts,  EBCF  and 
AEFD,  to  each  other  as  m  to  n. 


Suppose  the  sides  includmg  the 
/  \      parallel  to  be  produced  to  meet 

~  at  0.    The  triangle  BOC  may  be 

regarded  as  known.    Call  its  area 
a.    The  trapezium  EBCF  is  known  as  to  area,  being 

m  X  ABCD 

.    Call  this  area  6. 

m  -f  n 

The  area  of  the  triangle  AOD  is  known.  Call  it  c.  Its 
side  AO  is  also  known. 

Now,  (Art.  19,  Prob.  1,  Case  2),  OE  =  AO 

Whence,  BE  =  OE  —  OB. 

Measure  this  distance  and  run  EF  parallel  to  AD. 

Another  procedure  is  to  draw  .57  parallel  to  AD,  form- 
ing the  triangle  BCI,  whose  area  and  side  BI  may  be 
found;  whence  the  ratio  of  the  trapezoid  EBIF  to  the 
trapezoid  ABID  is  obtainable,  and  accordingly  the  dis- 
tance BE.  - 

Sen.— The  problem  of  parting  off  a  given  area  from  a  trapezium  by 
a  line  parallel  to  a  side,  is  essentially  the  same  as  the  above. 

Examples. — 1.  The  field  being  as  given  in  Ex.  1,  Case  1, 
it  is  required  to  find  the  measurement  for  locating  a  par- 
allel to  BC  that  shall  divide  the  field  into  two  parts  in 
the  ratio  of  3  to  4. 

2.  Find  the  measurement  required  to  part  off  from  the 
same  field  an  area  of  4. 5  A.,  by  a  line  parallel  to  BC. 


Y 


162  A  MANUAL   OF  LAND   SURVEYING. 

22.   Prob.  4.    Two  men  own  land  situated  between  a 
.     .  road  XX'  and  a 

line  YY',  and  di- 
vided by  a  line 
BA'. 

It  is  required  to 
run  a  line  AB',  at    . 
right  angles  with 
the   road,   which 
TTT-    shall  part  off  areas 


'—    of  equal  value  from 
FlG<  61-  the  two  portions. 

Solution.— Let  T  be  the  triangle  AOB,  and  T'  the  tri- 
angle A'OBf. 

Let  v  ==  value  per  acre  of  T,  and  v'  —  value  per  acre 
of  T'. 

Let  angle  OBA  =  B,  and  angle  OA'B'  =  A'  be  known; 
and  let  AB  =  x,  BA'  =  c,  and  BO  =  z. 

We  shall  then  have 

z2  sin  B  cos  B 

area  T  == ,  (1) 

2 

(c  —  2)2sin^'  cos  B 

and  area  T'  =  —  — ,         (2) 

2  cos  (A'  —  jS) 

By  conditions  of  the  problem,  3T0  =  Tfio'. 
Whence,  T  \  T'  \ :  v'  :  v.    Let  the  ratio  v'  :  v  =  r. 
Then  T  =  T^-.    Whence,  from  (1)  and  (2), 
z2  r  sin  J/ 


(c  —  zY      sin  ^  cos  ( A'  —  .B) 

rsin^/  }* 


=  ?*. 


c  —  z        (  sin  I?  cos  (J/  —  .B)  J 

c»  en  cos  5 

Whence,  £  = ,  and  x  =  z  cos  A=—       — . 

n-\-  1  n  +  l 


COMPUTING   AEEAS. 


163 


23.  Many  problems  which  the  surveyor  meets  with 
may  be  readily  solved  by  trial  lines  and  successive 
approximations.  A  line  is  run  or  assumed  to  meet  the 
required  conditions  as  nearly  as  can  be  judged.  The 
area  parted  off  by  the  line  is  computed  and  the  amount 
of  error  found.  A  new  line  is  assumed  to  correct  the 
error,  and  thus  successive  approximations  to  the  true  line 
are  made  until  the  error  disappears.  If  good  judgment 
is  used,  it  is  sometimes  the  quickest  and  easiest  method 
to  solve  the  problem.  » 

Example. — The  northwest  quarter  of  Section  30  is  di- 
vided by  an  angling  road.  The  owner  wishes  it  laid  off 
into  five  acre  lots,  commencing  at  the  south  end,  the  lot 
lines  to  be  parallel  with  the  quarter  line,  and  running 
from  the  center  of  the  road  west  to  the  section  line.  Re- 
quired the  number  of  lots,  the  area  of  the  fractional  lot, 
if  any,  at  the  north  end,  and  the  dimensions  of  the  several 
lots.  The  total  dimensions  are  given  on  the  figure. 


5-ZCtl.OTi.  I 


23.38ch. 


Solution,  (First  Lot).— 
Length  of  south  line,  7.65 
ch.  If  the  lot  were  a  rect- 
angle of  9.00  chains  base, 
the  perpendicular  ac  would 
50.0000 

be  =  5.555-1-  chains. 

9.00 

Assume  that  ac  =  5.60,  to 
find  cd.     The  line  bx  di- 
verges from  ay  at  the  rate 
23.38  —  7.65 

of or  .39325  ch. 

40.00 

per  chain.     Then  cd  = 
7.65  +  (560  X  .39325)  =  9.852 
chains.        Area  abed  = 
9.852    -f     7.65 
X    5.60    = 


FIG.  62. 


4.600056  acres.    This  is  too 
small  by  .1  acre,  which  must  be  added. 
11 


164  A  MANUAL   OF  LAND   SURVEYING. 

For  the  next  approximation  we  observe  that  the  addi- 
tion of  1  link  along  the  line  cd  adds  nearly  .01  A.  to  the 
area.  So  we  will  add  10  links  for  the  trial.  5.60  -f  .10  = 
5.70,  and  5.70  X  .39325  =  2.2415  =  divergence  of  lines. 
2.2415  +  7.65  +  7.65 

X  5.70  =  4.99933  A.     The  result  is 

2 

still  a  trifle  short,  but  in  ordinary  surveying  would  be 
sufficiently  correct. 

To  find  the  remaining  side  rff  the  lot,  bd,  we  have  a 
right  triangle,  with  a  base  equal  to  ac  and  perpendicular 
equal  to  cd  —  «6. 

The  method  is  now  sufficiently  described  so  that  the 
student  may  finish  the  computations  and  make  a  plat  of 
the  example. 

Field  Notes.— Nearly  every  surveyor  has  a  method 
of  his  own  for  keeping  the  field  notes  of  his  surveys. 
For  general  purposes  probably  no  better  plan  has  been 
devised  than  that  employed  in  the  United  States  land 
surveys.  This  method  gives,  in  a  condensed  narrative 
form,  each  item  in  the  survey,  in  the  order  in  which  it 
was  executed,  and  affords  opportunity  for  explaining  all 
the  details  as  fully  as  may  be  .necessary. 

It  is  a  common  fault  among  surveyors  to  condense 
their  notes  into  the  least  possible  space  by  omitting  many 
things  of  importance  and  by  the  use  of  arbitrary  signs, 
which  may  or  may  not  be  understood  by  any  one  else  who 
may  have  need  to  refer  to  them.  The  notes  are  thus  de- 
prived of  much  of  their  value,  and  in  case  it  were  desired 
to  use  them  as  evidence  in  the  courts,  they  might  be 
excluded  altogether. 

The  field  notes  should  be  full  and  explicit,  and,  espec- 
ially in  re-surveys,  should  state  in  plain,  concise  words 
every  material  fact  in  regard  to  the  work  done.  Starting 
points  should  be  described  and  identified;  the  direction 
of  lines,  how  determined,  whether  from  the  true  merid- 
ian, the  magnetic  meridian,  or  from  an  arbitrary  meridian 


FIELD    NOTES.  165 

r 

adopted  for  the  line,  should  be  shown,  it  is  not  enough 
to  say  that  the  survey  started  from  a  certain  corner. 
That  may  be  disputed,  and  the  notes  should  give  the 
evidence  by  which  it  is  known  to  be  the  corner.  Tell 
what  was  found  to  mark  the  corner.  If  a  bearing  tree  of 
a  former  survey  is  found,  give  its  direction  and  distance 
from  the  corner.  Make  everything  so  clear  and  plain 
that  the  average  citizen  can  understand  it  and  judge  of 
the  trustworthiness  of  the  survey.  The  following  is  a 
sample  of  the  field  notes  of  the  United  States  survey.  It 
is  an  extract  from  the 


FIELD       NOTES 


OK  THK  SURVKY  OF  THE 


SUBDIVISION  AND  MEANDER  LINES 


Township  No.  6  North,  Range  No.  34  East 

OF  THE 
PRINCIPAL  BASE  AND  MERIDIAN 

OK 

MONTANA  TERRITORY, 
as  surveyed  by 

WALTER    W.    DE  LACY, 
U.  S.  Deputy  Surveyor, 

Under  his  Contract, 

No.  87, 
Dated  July  3,  1830 


166  A   MANUAL   OF    LAND    SURVEYING. 

T.  6  K,  R.  34  E. 


Preliminary  to  commencing  this  survey,  I  ran  west  on  a  blank 
line  on  the  south  boundary  of  Sec.  36,  and  at  39.97  chs.  found  the 
%  sec.  cor.  and  at  80.01  chs.  found  the  sec.  cor.  As  the  east 
boundary  of  Sec.  31  crosses  the  Yellowstone  River  it  was  not  re- 
run. My  compass  will  therefore  run  the  same  line  as  the  exte- 
rior boundaries,  and  the  chaining  practically  agrees. 

Survey  commenced  August  6th,  1879,  with  a  Hurt's  improved 
solar  compass. 

I  commenced  at  the  cor.  to  Sees.  1, 2,  35,  and  36,  on  the  south 
boundary,  which  is  a  sandstone  30x8x2^  ins.  firmly  set  in  the 
ground,  with  one  notch  on  E.  and  5  notches  on  W.  edges,  and 
pits  18x18x12  ins.  in  each  sec.  5l/2  ft.  dist.  with  mound  of  earth 
2  ft.  high,  <iL/z  ft.  base  alongside.  Thence  I  run  North  bet.  Sees. 
35  and  36.  Va.  18°  3(X  E. 

Enter  scattering  timber.    Alexander's  house  bears  N.  31°  W. 
Leave  scattering  timber. 

Set  a  post  3  ft.  long,  3  ins.  square,  with  marked  stone,  12  ins.  in 
the  ground,  for  M  sec.  cor.,  marked  M  S.  on  W.  side,  dug 
pits  18x18x12  ins.  N.  and  S.  of  post  5*4  ft.  dist.,  and  raised  a 
mound  of  earth  \yz  ft.  high,  3l/2  ft.  base,  around  post. 
Alexander's  house  bears  S.  53%°  W. 
Enter  brush. 

Right  bank  of  the  Yellowstone  River.    Set  a  post  4  ft.  long,  4 
ins.  square,  with  marked  stone,  12  ins.  in  the  ground,  for 
meander  cor.  to  fractional  sees.  35  and  36,  marked  M.  C.,  and 
T.  6  N.  on  S., 
R.  34  E.  S.  36  on  E..  and 

S.  35  on  W  faces,  dug  pit  3  ft.  square,  12  ins.  deep,  8  Iks. 
S.  of  post,  and  raised  mound  of  earth  2  ft.  high,  4yz 
ft.  base,  around  post. 

There  being  an  island  on  line  on  N.  side  of  channel,  I  send  a 
flag  across,  and  set  it  on  line  bet.  sees.  35  and  36,  on  bar  S. 
of  island.  I  then  go  across  to  flag  and  run  a  base  line  W. 
H.14  chs.,  to  a  point  from  which  meander  cor.  on  right  bank 
bears  S.  37°50'E.,  which  gives  for  distance  across  the  river 
to  edge  of  bar  14.34  chs.  I  then  run  north  from  flag  66  Iks. 
to  south  bank  of  island,  making  the  whole  distance  53.82  + 
14.34 -f  0.66  chs.,  or 

To  south  bank  of  island,  which  point  I  established  by  setting  a 
post  4  ft.  long,  4  ins.  square,  with  marked  stone,  12  ins.  in 
the  ground,  for  meander  cor.  to  fractional  sees.  35  and  36  on 
S.  bank  of  island,  marked  M.  C.,  and 
T.  6  N.  on  N., 
R.  34  E.  S.  36  on  E.,  and 

S.  35  on  W.  faces,  dug  pit  3  ft.  square,  12  ins.  deep,  8  Iks. 
N.  of  post,  and  raised  a  mound  of  earth  2  ft.  high, 
4V£  ft.  oase,  around  post. 

Thence  continue  on  line  across  island,  enter  brush. 
Leave  brush,  enter  timber. 

Set  a  post  4  ft.  long,  4  ins.  square,  with  marked  stone,  12  ins.  in 
the  ground,  for  cor.  to  sees.  25, 26, 35,  and  36,  marked 
T.  6N.  S.25onN.  E., 
R.  34  E.  S.  36  on  S.  E., 
S.  35  on  S.  W.,  and 
S.  26  on  N.  W.  faces,  with  1  notch  on  S.  and  E.  edges, 

from  which 
A  cottonwood,  12  ins.  diam.,  bears  N.  12%°  E.,  180  Iks. 

dist.,  marked  T.  6  N.,  R.  34  E.,  S.  25  B.  T. 
A  cottonwood,  18  ins.  diam.,  bears  S.  82°  E.,  154  Iks. 

dist.,  marked  T.  6  N  ,  R.  34  E.,  S.  36  B.  T. 
A  cottonwood,  10  ins.  diam.,  bears  S.  291/2°  W.,  56  Iks. 

dist.,  marked  T.  6  N.,  R.  34  E.,  S.  35  B.  T. 
A  cottonwood,  10  ins.  diam.,  bears  N.  46y2°  W.,  119  Iks. 

dist.,  marked  T.  6  N.,  R.  34  E.,  S.  26  B.  T. 
Land,  level. 

Soil,  rich  loam— 1st  rate. 
Timber,  cottonwood  and  willow,  undergrowth  same,  12.30  ch. 


FIELD    NOTES. 


1C 


The  following  is  a  sample  of   Field    Notes  of  a 
Resurvey,  kept  upon  the  same  plan: 


SURVEY  ox  SECTION  14,  TOWNSHIP  2  SOUTH   RANGK  10  WEST, 

For  J.  R.  Comings  and  H.  Rowland. 
May  22, 1874. 

C.  Rowland*,67'  !'  CTMrfnmen.  S.  Comings,  Flagman, 

Commenced  at  the  S.  E.  corner  of  Section  14.  Found  a  piece  of 
strap  railroad  iron  driven  for  the  corner,  which  Hugh  Shaffer  says  he 
knows  to  have  been  kept  in  the  same  place,  unquestioned,  as  the  corner 
for  over  30  years.  Marked 

a  maple,       8  in.  diarn.,  S.  4.n°  W.,  77  Iks.  dist. 

a  burr  oak,  12  "       "      N.43°W.,123  " 


Chains. 


40.00 
80.24 


I  set  up  a  tall  flag  on  the  corner  and  then  ran  west  on  random 
Va.  2°  15'  E.,  setting  temporary  stakes  every  10  chains  in  line. 
Quarter  section  corner  lost. 


40.1. 


60.18 


having  surveyor's  mark  distinct  on  it.    Set  a  piece  of  steel 

T  rail  28  inches  long  for  corner.    Marked 

locust,      16  in.  diam.,  S.  28°  W.  ,116  Iks.  distant, 
burr  oak,  18  "       "      N.  78°  E.,  152   "        " 

Ran  thence  east  on  corrected  line  at  single  sight  with  transit, 

from  corner  to  corner.       Va.  2°33'E.       10:30  A.M.. 
Found  cedar  stake  3  feet  below  surface  of  road  crossing  and  2V£ 

links  south  of  line.  No  other  evidence  of  corner  to  be  found. 

Put  a  piece  of  T  rail  24  inches  long  on  top  of  the  stake  for 

quar.  sec.  cor.,  55  links  south  of  south  rail  of  M.  C.  R.  R. 

No  tree  near. 
Planted  granite  boulder  20x12x6  inches,  with  cross  4-  mark,  for 

Va  quarter  corner,  in  true  line  between  qr.  post  and  section 

corner  and  marked 

maple,      12  in.  diam.,  S.  ]tf>  E.,  55  Iks.  distant, 
burr  oak,  16"        "      N.54JL.,  US    " 


In  some  surveys,  such  as  laying  out  additions  to  cities 
or  villages,  or  any  similar  work,  it  is  better  to  make  a 
rough  sketch  or  plat  of  the  work  in  the  field  book  and 
mark  the  dimensions  and  directions  cf  lines  on  the  plat. 
Field  bocks  which  are  ruled  in  small  cross  sections  are 
best  adapted  to  this  use. 

ABBREVIATIONS. —  Where  the  work  of  the  land  sur- 
veyor consists  in  re-surveys  and  sub-dividing  sections 
of  the  United  States  Surveys,  the  field  notes  may  be 
made  more  concise  and  liability  of  error  reduced  by  al- 
ways using  a  definite  symbol  to  refer  to  each  corner  of 
the  section  or  sub-division.  The  symbols  should  be 
simple  and  adopted  upon  some  system  by  which 


168 


A    MANUAL,    OF    LAND    SURVEYING. 


10 


they  may  be  easily  remembered  and  located  in  trie 

mind. 
The    system    shown    in    the    figure    has  been    used 

many  years  by  surveyors  in  Michigan  and  found  sat 
isfactory. 

All  the  corners  lying  in  the 
exterior  lines  of  the  section  are 
numbered  in  a  definite  order  of 
rotation  in  accordance  with 
their  relative  importance.  Let- 
ters are  used  for  the  interior 
corners,  the  first  letters  being 
used  for  the  corners  lying  in  the 
quarter  lines  and  the  others  for 

the  centers  of  the  quarter  sections. 
The  following  is  a  sample  of  the  manner  of  using  the 

symbols  in  keeping  notes  upon  the  U.  S.  System  when 

sub-dividing  a  section. 

Began  at  7.    Found  stake  in  place  and  both  bearing  trees  stand- 
.  ing.   Planted  stone  25"  X  8"  X  6"  marked  -+-  for  corner.   Thence 
north  on   random.     Var.  2°  3(X  E,  setting    temporary  stakes 
every  10  chains 

Intersected  Section  line  26  links  west  of  5. 
At  5  found  rotten  stake  at  correct  point,  S.  28°  W.  66  Iks  from  stump 

of  W.  Oak  bearing  tree  of  U.  S.  Survey. 

Drove  stake  for  corner  and  put  broken  earthenware  and  glass 
around  it  and  marked 

Wh.  Oak  12"  d  ;  N.  66°  E.  42  Iks. 
Wh.  OnklS          N   34  W.  63  Iks. 


9          5        10          < 

e 

a 

t 

d 

b 

\ 

h 

t 

c 

g 

13 


).22 


39.92 


9.98 
19.96 


29.94 


20.02 
40.18 
80.04 


From  5  ran  east  on  random,  setting  temporary  stakes   every  10 

chains. 
Intersected  Sec.  line  12  Iks.  North  of  2.    Found  earthen  post  in 

correct  position    and    bearing    trees   of    resurvey  standing. 

Thence  West  on  corrected  line. 
Set  stake  on  true  line. 

At  10  set  stake  with  stones  around  it  and  marked 
Pine  12  N.  46°  W.  79  Iks.  dist. 
Red  Oak  24  S.  19£°  W.  72  dist. 
Set  stake  on  true  line. 


From  10  ran  south  on  random  Var.  2°  19'  E.  and  set  temporary 
stakes  at  20  and  40  chains. 


Then  went  to  6.  Found  post  and  bearing  trees  of  resurvey  stand- 
ing. Ran  thence  West  on  random  Var.  2°  20'  E. 

Intersected  random  from  North  6  links  South  of  temp,  stake. 

Intersected  random  14  line  8  links  North  of  temp,  stake. 

Int.  Sec.  line  10  links  South  of  8.  Corner  post  dug  out  in  road. 
Set  iron  plow  beam  for  corner  S.  29  W.  76  Iks.  from  bearing 
tree  of  U.  S.  Survey. 

Thence  East  Corrected  line. 

At  intersection  of  quarter  lines  set  post. 


FIELD    NOTES. 


169 


The  following  method  of  abridging  field  notes  is  used 
by  the  land  department  of  the  United  States.  The 
plat  of  a  township  is  lettered  and  numbered  as  shown 
in  the  diagram.  Corners  in  the  township  boundary  are 

(r     s     - 


9 
H 

J 
i 

K 

A 

L 

I 

0 

**»* 

tffc. 

4t 

**«• 

T 

2 

*<*><• 

Y 

X 
(( 

f 

s 

3 

2 

7 

s 

fi 

5 

W 

11 

f     1 

2 

6 

s 

# 

i 

1 

8 

*    J 

7 

6 

*     / 

s 

y 

1 

\ 

3 

2 

W 

I, 
? 

i 

9 

j    <? 

0 

3       2 

j 

3       ^ 

2 

3        2 

.T 

?      2 

4 

6 

i 

3 

o 

?•   2 

g 

?       ? 

ft 

? 

2      2 

6 

3       2 

f 

6 

s 

3 

g 

V 

,r 

/ 

# 

?    3 

3 

3       3 

4- 

•f    '* 

f 

f     3 

6 

A 

r   *>   o    *  f>  P   c 

9    A 

?      r     S      $      T 

referred  to  by  letter;  e.  g.,  3  or  k.  Interior  section 
corners  are  referred  to  by  the  numbers  of  the  sections ; 
e.  g.y  corner  of  9,  10,  15,  and  16.  Interior  quarter  sec- 
tion corners  are  referred  to  by  their  position  on  the 
lines,  e.  0.,  K  to  W  at  3  or  E  to  G  at  6.  The  descrip- 
tions of  corners  thus  referred  to  are  writteniout  in  the 
margins  of  the  plats,  while  all  other  matter  contained 
in  the  field  notes  is,  as  far  as  possible  marked  on  the 
plats  themselves.  The  letters  along  the  margin  of  the 
diagram  are  arranged  the  same  as  in  Plate  III,  Instruc- 
tions of  1902.  A  different  arrangement  has  been  used 
commencing  in  the  upper  left  hand  corner  and  passing 
around  the  plat  in  the  opposite  direction. 


170  A  MANUAL   OF  LAND   SURVEYING. 


CHAPTER  VII. 


CTJRVELINEAR  SURVEYING. 

1.  As  land  surveyors  have  occasion  in  laying  out 
streets  in  villages,  parks,  cemeteries,  race  courses,  draims, 
etc.,  sometimes  to  make  use  of  curved  lines,  it  has  been 
deemed  proper  to  include  in  this  work  a  short  discussion 
of  the  manner  of  locating  the  simpler  curves,  and  add 
such  tables  as  are  needed  for  this  use.  For  a  more  com- 
plete exposition  of  the  subject,  consult  the  field  books  of 
Henck,  Trautwine,  Shunk,  or  Searles. 

The  curve  most  commonly  used  is  the  circular  curve, 
simple  or  compound. 

The  simple  circular  curve,  as  its  name  indicates,  is  a 
circle  or  an  arc.  When  an  arc  is  used  to  connect  two 
straight  lines,  these  lines,  from  their  relation  to  the  circle, 
are  termed  tangents. 

The  compound  circular  curve  is  a  combination  of  arcs 
having  different  radii.  At  the  point  of  junction  of  any 
two  of  these  arcs  their  radii  lie  in  the  same  straight  line. 

Of  the  several  geometrical  propositions  on  which  the 
theory  of  running  curved  lines  depends,  it  will  not  be 
necessary  for  our  purpose  to  recall  more  than  the  fol- 
lowing 

PRELIMINARY   PROPOSITIONS. 

1.  If  a  circle  be  drawn  touching  each  of  two  intersecting 
lines  at  but  a  single  point,  then  the  exterior  angle  made 
by  the  intersection  of  these  lines  is  equal  to  the  angle  at 
the  center  of  the  circle  which  is  measured  by  the  arc 
intercepted  by  the  two  lines  at  their  points  of  tangency. 

2.  The  angle  which  either  line  makes  with  the  chord  of 
the  intercepted  arc  equals  one-half  the  angle  at  the  centre 
of  the  circle  which  is  subtended  by  that  chord. 


CCTtVELINEAR    SURVEYING. 


171 


In  Fig.  63  CF  and 
TI  represent  the  two 
lines  tangent  to  the 
circle  at  C  and  f>aiid 
intersecting  at  I.  The 
angle  FIT=^Of^ 
and  the  angle  &CT^= 
Yz  COT. 


FIG.  63. 


The  angle 
called  the 
angle,  and  the?aiigle 
FCT  the  tangential 
angle. 


Curves  are  named  from  the  angle  which  is  subtended 
by  a  chord  ICC)  feet  long.  Thus,  if  the  100  foot  chord 
subtends  an  angle  of  1  degree,  the  curve  is  spoken  of  as 
a  1°  curve;  if  of  5°,  as  a  5°  curve,  and  so  on.  Tables  have 
been  prepared  giving  the  various  functions  of  a  1°  curve, 
which  are  of  great  assistance  in  running  curved  lines, 
saving  nearly  all  the  trouble  of  calculation.  The  foot  is 
taken  as  the  primary  unit  of  these  tables  and  is  most 
commonly  used,  but  any  other  unit  using  the  decimal 
notation,  as  a  link  or  metre,  is  just  as  readily  applied. 

Curves  are  run  on  the  ground  by  successive  deflections 
of  chords.  The  amount  of  each  deflection  may  be  meas- 
ured on  the  ground  with  the  tape  or  turned  off  on  the 
transit. 

2«        To  run  a  Curve  with  Pickets  and  Tape. 

—  First,  determine  the  radius  of  the  curve  and  the  length 
of  chord  to  be  used.  The  latter  is  usually  100.  From 
these  data  the  amount  of  deflection  for  each  chord  is 
determined  as  follows: 


chord2 

Defl.  dist.  =  -        Tangential  d:st. 
radius 


defl.  disk 


A  MANUAL  OF   LAND  SURVEYING. 


IlG.  64. 


Example  1. — Let  ab  be  the  straight  line  or  tangent 
which  is  to  be  continued  from  6  by  a  curve  having  a 
radius  of  1,433  feet,  using  chords  of  ICO  feet. 

Extend  the  line  ab  to  c,  making  be  =  i/bd2— cd2.-  Ex- 
tend the  chord  bd  to  e,  making  de  =  bd  =  df.  Extend 
the  chord  df  in  a  similar  manner,  cbd  is  the  tangential 
angle,  and  cd  the  amount  of  the  deflection  to  be  meas- 
ured from  the  tangent  to  find  the  line  of  the  curve,  edf 
is  the  deflection  angle,  and  ef  is  the  amount  of  deflection 
to  be  measured  off  from  the  extension  of  the  chord  bd  to 
find  the  line  of  the  curve. 

To  find  the  distance  ef.— The  triangles  edf  and-  dof 

df1  10G2 

being  similar,  ef  :  df  ::  df  •  do.  /.  ef=  —  == 

do  1433 

=  6.98  nearly.  The  tangential  deflection  being  one-half 
the  chord  deflection,  cd  =  y^ef  =  3.49.  The  triangle  bed 
is  right-angled  at  c,  hence  be  —  ybd*  —  cd2  =  /lOO2  —  3.492 
=  99.94.  The  difference  between  be  and  bd  is  so  small 
that  in  all  curves  of  large  radius  it  may  be  neglected  on 
the  ground  and  be  be  measured  oft'  =  bd. 

These  lines  may  be  run  with  pickets,  the  chords  meas- 
ured with  the  tape,  and  the  deflections  when  not  too  large 
measured  off  by  a  graduated  rod  or  a  rod  cut  to  the  exact 
length. 


CTJRVELIXEAR  SURVEYING.  1-7  3 

Example  2.— Lay  off  on  the  ground  a  curve  having  a 
radius  of  2,640  feet,  using  chords  of  50  feet.  \  ?  '' 

Ex.  3  —Lay  off  a  curve  having  a  radius  of  819  feet  and 
chord  of  50  feet.  '-  ."  „ 

Ex.  4.— Lay  off  a  curve  with  radius  2,865  feet,  chord  100 
feet. 

Ex.  5. — Lay  off  a  curve  with  radius  1,910,  chord  100. 

j£Xm  £. — Lay  off  a  curve  with  radius  882,  chord  50r 

Ex.  7.— Lay  off  a  curve  with  radius  1,042,  chord  X00. 

*       i 

3.  Keeping  the  Field  Notes  of  Transit 
Lines.— The  field  notes  of  transit  work  where  long  line's 
are  being  run,  as  for  railroads,  drains,  etc.,  are  usually 
kept  in  a  different  manner  from  those  of  other  surveys. 
The  notes  proper  are  kept  on  the  left-hand  page  of  the 
field  book.  The  opposite  page  is  used  for  explanatory 
matter,  sketches  of  topography  along  the  line,  such  as 
road  and  stream  crossings  and  obstacles  in  line,  in  greater 
or  less  minuteness  of  detail  according  to  circumstances. 
The  line  is  marked  by  stakes  driven  at  regular  intervals, 
usually  100  feet  or  100  links,  and  numbered  from  0  up- 
wards. The  corresponding  numbers  are  kept  on  the  left- 
hand  column  of  the  note  book,  commencing  at  the  bottom 
of  the  page  and  running  upwards. 

If  the  topography  is  sketched  on  the  right-hand  page, 
th$  number  of  every  stake  is  put*  down  in  its  regular 
order,  and  the  ruling  of  the  book  forms  a  scale  by  which 
the  sketches  are  made.  A  book  ruled  in  cross-sections  is 
very  convenient  for  this  work.  If  the  topography  is  not 
taken,  the  important  stations  are  noted  down  and  the 
intermediates  are  omitted.  The  following  abbreviations 
are  used:  P.  I.,  point  of  intersection;  P.  C.,  point  of  curve, 
or  point  where  the  curve  begins;  P.  C.  C.,  point  of  com- 
pound curve;  P.  R.  C.,  j>oint  of  reverse  curve;  P.  T.,  point 
tangent,  or  point  where  the  curve  ends;  T.  P.,  turning 
point,  indicating  where  the  transit  was  set  up,  also  indi- 
cated by  O  or  /\. 

The  direction  of  the  tangents  is  kept  as  shown  by  the 
magnetic  needle.  This  serves  as  a  check  on  the  angles  of 
deflection,  and  assists  in  locating  errors. 


174  A  MANUA-L  OF  LAND   SURVEYING. 

SPECIMEN  OF  ABRIDGED  NOTES. 


[LEFT  PAGE.] 

Notes  of  Line  "B,"  D.  &  R.  G. 
mile  above  the  Dead  Horse 
satch  Co.,  Utah. 


[RIGHT  PAGE.] 

W.  R.  R.,  commencing  about  a 
Crossing  of  Price  River,  Wan- 


Sta. 

32°2(X/ 

\ 

300 
200 

P.  C.  4°  C 
P.  T.      S. 

urve  r't. 

80°  E. 

Def.  32°  20' 
15°  (XX 
7°30/ 

\ 

7 

19 

6°  00' 

\ 

18 

4°  30' 

-f  60  Old  Spanish 

\  Trail,  S.70°W. 

17 
16 

3°  (XX 
1°30' 

\15° 

15  o 

P.  C.  3°  C 

urvel'ft 

0 

P.I.  at  17  +  51.4. 

10  o 

jf/ 

4 
3 

Indian  trail  / 

2 
1 

+50  to  Wash^^ 

^.       20  ft.  wide, 
=    10  ft.  deep. 

Oo 

S.  65°  E. 

4.     To  Run  a  Curve  with  the  Transit.— The 

transit  is  set  up  on  the  point  in  the  tangent  from  which 
the  curve  is  to  commence.  The  limb  is  clamped  with  the 
verniers  at  zero,  the  telescope  ranged  along  the  line  of 
the  tangent,  and  the  instrument  clamped  in  that  position. 
The  tangential  angle,  =  %  the  deflection  angle,  is  then 
turned  off  on  the  limb.  The  leading  chain-man  draws  out 
the  chain  or  tape  in  the  desired  direction,  and  when  out 
at  full  length,  places  his  rod  in  line  as  directed  by  the 
signals  of  the  transit-man.  He  then  carefully  measures 
the  length  of  the  chord,  marking  the  distance  with  his 
rod,  which  is  then  aligned  the  second  time.  A  stake  is 
driven  to  mark  the  point,  and  the  chain-men  go  ahead 
and  measure  the  second  chord,  being  aligned  by  the 
transit-man  as  before,  and  thus  continue  as  far  as  neces- 
sary or  convenient.  The  transit-man  turns  off  equal 


CUBYELINEAB  SURVEYING.          175 

angles  on  the  transit  for  each  successive  chord  as  it  is 
measured.  At  the  end  of  the  last  chord  which  is  run 
from  any  one  setting  of  the  transit,  a  short  stake  is 
driven  firmly  into  the  ground  and  a  tack  driven  in  the 
top  of  the  stake,  to  mark  the  exact  point.  If  the  curve 
is  to  be  continued,  the  transit  is  moved  up  to  this  point, 
and  with  the  limb  clamped  as  it  was  used  at  the  last 
observation,  the  telescope  is  ranged  back  to  the  point 
from  which  the  observation  was  taken,  and  the  instru- 
ment clamped  in  that  position.  As  the  angles  have  all 
been  turned  off  from  a  point  in  the  circumference  of  the 
circle,  they  are  only  half  as  great  as  the  angle  at  the 
center  subtended  by  the  same  chords.  Hence  the  transit- 
man  now  unclamps  the  limb  and  turns  off  as  much  more 
angle  as  he  had  previously  laid  off.  This  gives  him  a  new 
line,  tangent  to  the  curve,  from  which  he  may  continue 
to  lay  off  chords  as  before. 

Some  transit  men,  instead  of  doubling  the  angle  after 
the  back-sight  is  taken,  turn  off  an  equal  amount  in  the 
opposite  direction  on  the  limb  before  taking  the  back- 
sight. Then,  after  getting  the  back-sight,  the  verniers  are 
brought  to  zero  on  the  limb,  when  the  telescope  will  give 
the  line  of  the  new  tangent,  as  before. 

Curves  are  usually  run  to  connect  two  straight  lines 
which  have  been  previously  located.  In  such  a  case,  pre- 
.liminary  to  running  the  curve,  it  is  necessary  to  find — 

1st.     The  deflection  angle  between  the  lines. 

2nd.    The  radius  of  the  curve  to  be  used. 

3d.      The  P.  C.  and  P.  T. 

4th.     The  length  of  the  curve. 

The  manner  of  procedure  in  such  a  case  is  indicated  in 
the  following: 

Example  1. — To  join  two  straight  lines  having  a  deflec- 
tion angle  of  48°  16',  by  a  curve  the  middle  point  (/)  of 
which  shall  be  at  a  distance  of  112  feet  from  the  point  of 
intersection. 

Assume  that  the  line  abn  has  been  marked  with  stakes 
100  feet  apart,  and  that  the  point  of  intersection  is  found 
to  be  at  stake  No.  116,  -f  43.7;  in  other  words,  that  the 
P.  I.  is  43.7  feet  beyond  stake  No.  116. 


176 


A   MANUAL   OF   LAND  SURVEYING. 

PC  Pl 


FIG.  65. 

The  transit  is  set  up  over  the  point  of  intersection,  the 
verniers  clamped  at  zero,  the  telescope  reversed  and. 
ranged  along  the  line  ab,  and  the  instrument  clamp  3d  in 
that  position.  The  telescope  is  then  righted,  the  appsr 
clamp  loosened,  the  telescope  turned  and  the  limb  again 
clamped  with  the  telescope  pointing  along  the  line  cde, 
and  the  angle  read  =  48°  16'.  Before  proceeding  further, 
it  is  necessary  to  determine  the  degree  of  curve  to  be 
used.  By  the  conditions  of  the  example,  the  middle 
point  of  the  curve  is  to  be  112  feet  from  the  P.  1.  Turn- 
ing to  the  table  of  functions  of  a  1°  curve,  we  find  its 
external  secant,  cf\  to  be  548.8  feet  for  an  angle  of  48°  16'. 

548.8 

Dividing  this  by  112,  we  find =  4.9,  or  4°  54',  to  be 

112 

the  degree  of  curvature  to  be  used.  Next  we  find  the 
distance  be  =  cd,  which  is  to  be  measured  along  the  lines 
to  fincl  the  beginning  and  end  of  the  curve,  the  P.  C.  and 
P.  T.  Referring  again  to  our  table,  we  find  that  the 
tangent  of  a  1°  degree  curve  for  a  deflection  of  48°  16' 
is  2567.1,  which  divided  by  4.9,  the  degree  of  curvature, 
gives  523.9. 

We  now  measure  from  the  P.  I.  523.9  feet  along  the  line 
cde,  and  set  a  hub  and  drive  a  tack  in  it  for  the  P.  T.  In  a 
similar  manner  we  next  locate  the  beginning  of  the  curve, 
which,  subtracting  5  -f  23.9  from  116  -f  43.7,  we  find  to  be 
at  Station  111,  +  19.8.  If  the  ground  be  clear  and  open, 


-CUKYELINEAR  SURVEYING.  177 

so  that  thE  wflole  curve  may  be  seen  at  once,  the  transit 
may  now  be  set  up  on  the  P.  T.,  and  the  whole  curve  and 
as  much  of  the  next  tangent  cle  as  desired  run  at  one 
setting  of  the  instrument,  at  the  same  time  avoiding 
most  of  the  errors  usually  made  in  running  the  curve 
from  the  P.  C.  If  this  cannot  be  done,  the  transit  is  set 
up  at  the  P.  C.  with  verniers  at  zero  and  a  foresight  on 
the  P.  I.,  or  back-sight  to  some  point  along  the  line  ab. 
The  P.  C.  being  at  Sta.  Ill,  +  19.8,  the  first  deflection  will 
be  for  the  partial  chord  found  by  subtracting  19.8  from 
100  =  80.2,  or  .802  of  the  full  deflection.  The  tangential 
deflection  for  a  full  chord  being  2°  27',  for  the  partial 
chord  would  be  .802  of  2°  27'  =  1°  58'  for  the  first  deflec- 
tion. For  each  subsequent  full  chord  2°  27'  additional  is 
turned  off  on  the  transit  as 'far  as  the  line  can  be  seen. 
Say  that  the  line  cannot  be  seen  farther  than  Sta.  116;  the 
several  deflections  up  to  that  point  would  be,  for  Sta.  112, 
1°  58';  Sta.  113,  4°  25';  Sta.  114,  6°  52';  Sta.  115,  9°  19';  Sta. 
116,  11°  46'.  A  hub  and  tack  are  driven  at  Sta.  116,  and 
the  transit  moved  up  to  that  point  or,  what  is  better, 
to  the  P.  T.,  if  the  station  is  visible  from  there.  If  the 
transit  is  set  up  at  Sta.  116,  the  back-sight  is  taken  on  the 
P.  C.,  with  the  limb  clamped  at  11°  46',  as  at  the  last 
observation.  The  telescope  is  then  righted,  and  an  addi- 
tional 11  C46'  turned  off  for  the  new  tangent,  from  which 
the  subsequent  deflections  are  turned  off.  For  Station 
117  the  deflection  would  be  11°  46'  +  11°  46'  -f  2°  27'  = 
25°  59';  for  Sta.  118,  28°  26';  for  Sta.  119,  30°  53';  for  Sta. 
120,  33°  20';  for  Sta.  121,  35°  47'. 

Before  passing  this  point,  we  must  know  the  length  of 
the  curve.  As  there  are  48°  16'  total  deflection,  and  each 
chord  cuts  off  4°  54'  of  it,  it  is  evident  there  are  as  many 
100  foot  chords  as  4°  54'  is  contained  in  48°  16'.  Reducing 

48.266 

the  minutes  to  decimals  and  dividing,  we  have = 

4.9 

9.85  chords  for  the  length  of  the  curve.  This  added  to 
111  -f  19.8  gives  us  121  -{-  04.8  for  the  end  of  the  curve, 
and  04.8  feet  for  the  last  partial  chord.  We  find  the 

32 


178  A   MANUAL  OF  LAND  SURVEYING. 

deflection  for  this  distance  to  be  .07',  giving  for  the  last 
deflection  35°  47'  -;-  07'  =  35°  54'. 

The  work  should  now  prove  itself,  by  coming  out  at 
the  stake  which  was  previously  set  for  the  end  of  the 
curve,  and  we  may  further  test  it  by  setting  the  transit 
up  at  the  P.  T.,  back-sight  to  Sta.  116,  with  the  instrument 
clamped  at  35°  54',  as  last  used.  Unclamp  the  limb  and 
turn  off  as  much  more  as  has  been  turned  from  Sta.  116, 
35°  54'  —  23°  32'  =  12°  22',  which  added  to  35°  54'  =  48°  16', 
the  total  deflection.  If  the  work  has  been  accurately 
performed,  a  back-sight  through  the  telescope  should 
strike  the  P.  I.  It  is  very  seldom  that  curves  run  in  this 
way  will  come  out  just  right,  hence  it  is  better  to  never 
set  up  the  transit  at  points  in  the  curve  between  the  P.  C. 
and  P.  T.  when  it  can  readily  be  avoided.  Still  it  is  the 
ordinary  and  sometimes  the  only  way  the  curves  can  be 
run. 

Let  the  student  make  the  necessary  calculations  to 
locate  curves  from  the  following  data : 

Ex.  2.— Total  deflection,  26°  50'.  External  (cf,  Fig.  65), 
120.87  feet.  P.  C.  at  Sta.  112,  +  40.  Transit  moved  every 
550  feet. 

Ex.  3.— Total  deflection,  35°  15'.  External,  126.2  feet. 
P.  I.  at  Sta.  262,  +  07.3.  T.  P.  at  Sta.  263. 

Ex.  4.— Total  deflection,  18°  36'.  Curve,  1°  2',  P.  I.  at 
96,  +  42.6.  T.  P.  at  Sta.  93  and  100. 

The  starting  point  of  a  curve  is  sometimes  so  situated 
that  it  is  not  convenient  to  set  up  the  transit  at  that 
point,  or  to  run  the  line  from  it  if  it  were,  as  in  streams,, 
gullies,  etc.,  and  it  then  becomes  convenient  to  set  up 
the  transit  first  at  some  intermediate  point  in  the  curve 
which  has  to  be  found. 

5.  To  Locate  a  Curve  from  the  Middle 
Point.— Set  the  transit  up  at  the  P.  I.  Bisect  the  inte- 
rior angle  bed  (Fig.  65 ).  Find  the  external  cf  of  the 
desired  curve  and  measure  it  off  on  the  line  of  bisection. 
This  gives  the  middle  point  of  the  curve.  The  transit  is 
then  set  up  at  this  point  and  a  back-sight  taken  either  on 
the  P.  C.  or  P.  I.,  and  the  curve  run  in.  Let  the  student 
make  the  necessary  calculations  and  give  the  various 


CTJRVELItfEAR    SURVEYING.  179 

deflections  which  would  be  used  on  the  transit  to  locate 
from  the  middle  point  the  curve  required  in  Ex.  1,  Fig. 
133,  the  first  back-sight  to  be  taken  from  the  P.  C.  Give 
the  same,  the  back-sight  being  taken  from  the  P.  I.  Also, 
solve  the  following  curves,  to  be  run  from  the  middle 
points,  back-sights  from  P.  C.,  also  from  P.  I.: 

Examples.— I.  Total  deflection,  16°  24'.  Curve,  1°  32'. 
P.  I.  at  96,  +  27. 

2-  Total  deflection,  26°  18'.  Curve,  2°  24'.  P.  I.  at  13, 
+  «2.7. 

3.  Total  deflection,  35°  40'.  Curve,  3°  16'.  P.  I.  at  97, 
-f  62.6. 

It  is   sometimes  convenient,  from  various  reasons — 

6.  To  Locate  the  Curve  with  the  Tran- 
sit at  some  other  Intermediate  Point  on  the 
Curve  than  the  middle.  Such  points  may  be  located 
by  ordinates  from  the  tangent.  This  is  usually  done  to 
avoid  obstacles  in  the  line  of  the  curve.  To  find  approx- 
imately on  the  ground  at  what  point  the  transit  may 
be  set  up,  the  following  formula  may  be  used: 

Let  x  =  length  of  the  ordinate, 

d  =  distance  along  the  tangent  from  the  P.  C., 
t  =  nat.  tangent  of  %  the  deflection  angle  of  the 
curve, 

Then  x  =  d?t. 

Example. — To  find  whether  the  transit  can  be  set  up  at 
a  point  on  a  4°  curve  opposite  a  point  on  the  tangent  4CO 
feet  from  the  P.  C. 

*  =  nat.  tang.,  2°  =  03.5.  d2  =  16.  /.  x  ==  56.  A  meas- 
ure of  56  feet  from  the  tangent  will  show  whether  the 
transit  can  be  set  up  at  this  point  or  not. 

It  will  be  fonnd  the  most  convenient  in  running  the 
curve  to  select  the  point  at  a  regular  station  at  the  end  of 
a  full  chord,  which  may  be  located  as  follows: 

Example  1.— Total  deflection,  48°  48'.  P.  1.  at  62,  -f  36. 
Curve,  4°  .  To  find  the  4th  full  station  on  the  line  of  the 
curve,  and  locate  the  remainder  of  the  curve  from  that 
point. 


180 


A   MANUAL   OF   LAND   SURVEYING. 


FIG.  66. 


First  find  the  number 
of    the   station    at    the 
P.  C.    be  =  tangt.  of  1° 
2599.2 

-4-  4  == =  649.8  or 

4 

6  +  49.8.  This  taken 
from  62  +  36  =  55  + 
86.2,  which  is  the  num- 
ber of  the  station  at 
the  P.  C.  From  here 
to  the  4th  fuli  station 
there  is  then  a  short 
chord  of  13.8  feet  and  four  full  chords.  The  tangential 
angle  cbd  is  therefore  4.138  X  2°  =  8°  16 J£';  whence  the 
deflection  angle  =  16°  33',  the  chord  of  which,  bd,  =  413.4. 
In  the  right-angled  triangle  bed,  we  now  have  the  side  bd 
=  413.4,  and  the  angle  cbd  =  8°  16^',  to  find  the  sides  be 
and  cd,  from  which  we  find  that  be  =•  409.1  and  cd  —  58.5. 
The  point  c  may  be  found  by  measuring  from  the  P.  I. 
649.8  —  409  =  240.7  =  ec.  Having  thus  located  the  point 
d,  which  is  Station  60  on  the  curve,  the  transit  is  set  up  at 
that  point,  with  the  vernier  clamped  at  90°  ,  and  a  back- 
sight taken  to  the  point  c.  The  upper  clamp  is  then 
loosened  and  the  limb  brought  to  16°  33',  which  gives  the 
tangent  from  which  the  remainder  of  the  curve  is  located. 
Let  the  student  calculate  the  following  curves: 

2.  Total  deflection,  36 '  20'.    P.  I.  at  26,  -f  44.6.    Curve, 
2°  30',  to  be  located  from  the  3rd  full  station  on  the  curve. 

3.  Total  deflection,  61°  18'.    P.  I.  at  42,  +  28.5.    Curve, 
4°  40',  to  be  located  from  6th  full  station  on  the  curve. 

4.  Total  deflection,  42°  50'.    P.  I.  at  112, -f  72.    Curve, 
3°  18',  to  be  located  from  Station  114  +  50  on  the  curve. 

7  .  Short  Curves.— When  the  deflections  between 
the  lines  are  but  small,  and  it  is  not  important  that  any 
particular  degree  of  curvature  be  used,  it  will  be  found 
convenient  to  make  the  curve  an  even  two  or  four  sta- 
tions in  length.  In  case  this  is  done,  the  curve  may  be 
marked  out  before  the  transit  is  moved  from  the  P.  I., 
after  observing  the  deflection  angle,  and  it  will  not  be 


CCRVELLSEAK  SUKVEYLSG. 


181 


necessary  to  set  it  up  on  the  curve  at  all.  The  middle  6f 
the  curve  will  be  located  by  laying  off  the  external  secant 
as  before  directed.  The  P.  C.  and  P.  I.  are  also  located  as 
usual.  If  four  stations  are  used,  the  intermediate  sta- 
tions may  be  determined  from  the  P.  I.,  the  same  as  if 
the  transit  were  at  the  P.  C.  or  P.  T.,  the  error  being  so 
small  that  it  may  usually  be  neglected. 

8-   Passing  Obstructions  in  the  Line.— One 

method  of  doing  this,  by  offset  from  the  tangent,  has 
already  been  sufficiently  explained.  Another  method, 
which  is  very  generally  applicable,  is  by  parallel  offsets 
from  the  curve.  An  offset  is  made  in  any  convenient 
direction  far  enough  to  pass  the  obstruction.  The  curve 
is  continued  from  this  point  till  the  obstacle  is  passed, 
when  the  true  line  is  regained  by  an  inset  equal  to  and 
parallel  with  the  offset.  If  the  lines  are  run  in  the  man- 
ner indicated  on  page  82,  (3),  this  will  be  a  very  simple 
matter,  as  the  telescope  will  always  point  in  the  same 
direction  when  the  verniers  mark  the  same  point  on 
the  limb. 


Fig.  67  illustrates  this  method 
of  passing  obstacles,  be  and  de 
are  equal  and  parallel. 


FIG.  67. 

9.  Compound  Curves,  being  a  combination  of 
simple  curves,  have  their  several  components  located  in 
the  same  manner.  They  are  usually  run  to  fit  the  topog- 
raphy of  the  country  through  which  they  are  laid,  in  order 
to  get  uniform  gradients  on  street  or  railroad  lines,  or 
save  labor  and  expense  in  construction. 


182 


A  MANUAL  OF  LAND  SURVEYING. 


*  Having  the  several  straight  lines  determined  which  are 
to  form  the  tangents  of  the  curve,  it  is  only  necessary  to 
find  the  degrees  of  curvature  of  the  several  component 
curves,  which  are  then  located  in  the  manner  already 
described.  Usually  there  will  be  found  on  the  ground 
special  reasons  for  selecting  a  particular  radius  for  one  of 
the  component  curves,  which  will  thus  dictate  the  radii 
of  the  rest. 

rl 


FIG.  68. 

Example  1. — Let  ac,  ce,  eg,  gi  and  ik  represent  tangents 
of  the  curve,  and  bed,  def,  fgJi  and  hik  the  angles  of 
deflection. 

Let  ce  =  1370,  eg  =  1200,  gi  =  1000. 

Let  bed  =  92°  ,  def=  36°  ,  fgh  =  233  15',  and  hik  = 
43°  30',  the  corresponding  curves  of  which  we  will  number 
1,  2,  3  and  4. 

Let  the  tangents  ce  and  eg  be  united  by  a  3^  curve. 

Required  the  radii  or  degrees  of  curvature  of  the 
remaining  components  of  the  curve,  and  the  length  of 
the  curve. 

SUGGESTIONS.— First  find  the  tangent  of  a  3°  curve  for 
an  angle  of  36°.  Tangent  of  1°  curve  for  36°  =  1861.8; 
.'.  for  3°  curve  =  620.6.  This  leaves  1370  —  620.6  =  749.4, 
length  of  tangent  of  curve  No.  1.  Tangent  of  1°  curve 
for  92°  =  5933.2,  which  divided  by  749.4  —  7.917°  or  7°  55', 
the  degree  of  curvature.  Radius,  724.3.  Length  of  curve 

92 
=  -   -  =  1162  feet.    We  find  the  tangent  of  curve  No.  3 

7.917 


CURVELINEAR  SURVEYING. 


183 


by  subtracting  the  tangent  of  curve  No.  2,  620.6,  from  the 
length  of  the  line  eg,  1200,  =  579.4.  The  tangent  of  a  1° 
curve  fur  a  dellection  of  23°  15'  we  find  from  the  table 
to  Le  1178.8,  which  divided  by  579.4  gives  the  degree  of 
curve  to  be  used,  2.034°  —  2"  02'.  The  calculations  for  the 
remainder  of  the  curve  are  made  in  a  similar  manner. 

It  is  customary  in  running  long  lines  for  drains,  rail- 
ways, etc.,  to  run  preliminary  lines  by  angles,  omitting 
the  curves,  till  the  location  of  the  tangents  is  definitely 
determined.  Stakes  are  set  and  numbered  the  same  as  on 
the  final  location.  Both  the  staking  and  measuring  are 
sometimes  omitted,  the  lines  being  run  as  simple  picket 
lines.  In  such  case,  when  the  final  location  is  made,  the 
line  is  staked  out  to  the  point  of  intersection  of  the 
tangents  and  afterward,  as  the  curve  is  run  in,  the  stakes 
between  the  P.  C.  and  P.  I.  are  taken  up  and  moved  to 
their  proper  place  in  the  line  of  the  curve. 

Examples  for  solution. — 1.  Let  the  student  calculate 
the  curves  and  plat  the  line  from  the  following  notes  of  a 
preliminary  angle  line,  making  all  the  calculations  that 
would  be  required  in  the  field,  and  giving  the  corrected 
numbers  of  the  stations  at  the  several  P.  C.'s,  P.  T.'s,  P. 
C.  C.'s  and  P.  R.  C.'s:— 


— 

176  0 

165  0 

P.I. 

Compound  Curve.    Angle  right,  14®. 

163  +  20  0 

P.I. 

Reverse  Curve.    Angle  right,  36°. 

153  0 

P.I. 

Reverse  Curve.    A  ng!e  left.  17°  26'. 

144  +  26  0 

P.I. 

2°  Curve.    Angle  right,  16°  3V 

129  0 

118  0 

108  0 

98+15  0 

P.I. 

of  Reverse  Curve.  Angle  right,  53°  12'. 

85  +  60  0 

P.I. 

Compound  Curve.    Angle  left,  16°. 

76  +  48  0 

P.  I. 

Compound  Curve.    Angle  left,  7C  8'. 

ffT     r^ 

P.  I. 

Cnrvr         \nf»lo  loft     l^ooo/ 

Of     o 

58  o 

48  o 

41  o 

P.I. 

23  16'  Curve.    Angle  right,  14°. 

30  o 

21  o 

P.I. 

3o  Curve.    Angle  left,  26°  32'. 

14  o 

0  o 

N.  45°  E. 

184  A  MANUAL  OF   LAND   SURVEYING. 

2.  The  following  are  notes  of  the  north  side  of  a  street  in 
Park  Beidler.  The  measures  are  taken  with  a  66  foot 
tape  of  100  links.  The  street  is  one  chain  wide.  A  tier 
of  lots  two  chains  deep  is  laid  out  on  each  side  of  the 
street.  The  lots  are  one  chain  wide  on  the  street,  and  are 
marked  by  stakes  set  and  numbered  at  regular  intervals 
of  one  chain.  The  lines  for  the  south  side  of  the  street 
and  for  the  back  ends  of  the  two  tiers  of  lots  are  to  be 
rim  with  the  transit  and  tape.  Required  the  details .  of 
these  lines  and  the  widths  of  lots  at  the  back  end,  the 
lot  lines  being  at  light  angles  with  the  street  and  on  the 
radii  of  the  curves. 


8  +  50 
6  +  20 


Intersect  west  line  of  Dawn  Street.     Course  M.  and  S. 

P.  T. 

P.  R.  C.    10°  Curve  right, 

P.  C.  C.    8°  Curve  left. 

P.  C.  C.    5°  Curve  left. 

P.O.    2°  Curve  left. 

P.  T. 

P.  C.  C.    6°  Curve  right. 

P.  B.  C.    4°  Curve  right. 

P.  C.  C.    4°  Curve  left. 

P.  B.  C.    8°  Curve  left. 

P.  C.  C.    4°  307  Curve  right. 

P.  C.    3°  Curve  right. 

East  at  right  angles  with  Sylvan  St.    Course  N.  and  S. 


The  following  formula  has  been  found  very  useful  in 
solving  many  problems  in  the  location  of  curves.  Like 
theformula(E  =  d2£  in  Art.  6,  it  is  designed  to  express 
%e  length  of  an  ordinate  from  the  tangent  to  the  curve: 

Let  x  =  length  of  the  ordinate, 

n  =  length  of  the  curve  in  chords  of  100  feet  each, 
d  =  degree  of  curvature. 

Then  x  =  ln2~d.  Thus  a  6°  curve  will  have  diverged 
from  its  tangent  at  the  end  of  500  feet,  |  X  52  X  6  =  131.25 
feet. 

By  making  d  equal  the  difference  of  the  degree  of 
curvature  of  two  curves  of  different  radii  but  having  a 
common  origin,  x  will  be  their  divergence  from  each  other 


OEIGIXAL    SURVEYS.  185 

at  the  end  of  n  stations.  This  formula  is  not  mathemat- 
ically exact,  and  therefore  gives  only  approximate  results, 
but  it  is  sufficiently  correct  for  all  ordinary  cases.  It  is 
easily  remembered;  it  requires  no  tables;  and  with  its  aid, 
with  such  modifications  as  a  little  ingenuity  will  suggest, 
and  a  table  of  actual  tangents  for  a  1°  curve,  the  surveyor 
can  solve  almost  any  case  that  will  ordinarily  arise  in  the 
field.  For  example:  Suppose  a  5°  curve  to  the  right  8 
stations  long  has  been  located,  and  its  extremity  falls  28 
feet  too  far  to  the  right  to  throw  the  tangent  on  the  best 
ground.  Making  x  =  28,  we  obtain  d  =  £,  showing  that 
a  4°  SO7  ctfrve  starting  from  the  same  origin  would  pass 
through  the  required  spot.  Again:  Suppose  that  in  this 
same  case  the  new  curve  is  to  commence  200  feet  back  of 
the  first  one;  then  the  required  divergence  from  the  tan- 
gent will  be  |  X  82  X  5  —  23  =  252.  Substituting  this 
value  for  x,  and  making  n  =  8  +  2,  we  have  d  =  2.88  = 
2°  53'. 


186  A  MANUAL   OF   LAND   SUKYEYIFG. 


CHAPTEE  VIII. 

OKIGINAL   SURYEYS. 

1.  In  land  surveying,  the  surveyor  has  two  distinct 
classes  of  problems  to  deal  with.    In  the  first  <jlass,  he  is 
called  upon 

(a)  To  lay  down  upon  the  ground  the  corners  and 
boundary  lines  of  tracts  of  land  of  specified  dimensions; 
and 

(6)  To  find  the  areas  of  tracts  which  are  already  defined 
by  natural  or  artificial  boundaries. 

In  this  class  is  included  the  original  marking  out  upon 
the  ground  of  the  boundaries  of  every  tract  of  land  how- 
ever great  or  small.  Hence  we  call  surveys  of  this  nature 
Original  Surveys. 

2.  When  tho  boundaries  have  once  been  laid  down 
upon  the  ground  an.*,  marked  by  persons  having  authority 
to  do  so,  then  the  surveyor,  who  is  afterward  called  upon, 
has  a  different  class  of  problems  to  deal  with.    He  then 
has 

(a)  To  find  the  corner  posts  and  monuments; 

(6)  To  re-locate  them  when  lost ;  and 

(c)  To  retrace  old  boundary  lines. 

Surveys  of  this  nature  we  shall  call  Resurveys. 

3.  Original  Surveys  include:  First.  The  rectangu- 
lar surveys  of  the  United  States,  known  as  the  govern- 
ment survey;  similar  surveys  in  Canada  and  other  coun- 
tries by  government  authority,  and  the  subdivision  of 
sections.    Second.  Surveys  made  by  the  proprietors  in 
those  regions  where  the  government  surveys  do  not  ex- 
tend, including  in  the  United  States  the  surveys  of  all 


ORIGINAL   SURVEYS.  187 

land  not  granted  by  the  original  states  of  the  Union  to 
the  general  government;  and  surveys  for  town  plats, 
highways  and  like  purposes. 

4  United  States  Survey.— The  territory  embraced 
within  the  present  States  of  Ohio,  Indiana,  Illinois,  Mich- 
igan, Wisconsin,  and  Tennessee,  that  part  of  Minnesota 
lying  east  of  the  Mississippi  Kiver,  and  all  of  Alabama 
and  Mississippi  lying  north  of  the  thirty-first  parallel, 
was  held  by  Massachusetts,  Connecticut,  New  York,  Vir- 
ginia, North  Carolina,  South  Carolina,  and  Georgia,  under 
grants  from  Great  Britain,  during  their  colonial  condi- 
tion. These  territorial  interests  were  surrendered  to  the 
General  Government  of  the  Union  by  the  last  named 
States  at  different  times  hereinafter  set  forth,  and  consti- 
tuted ttie  nucleus  of  our  public  domain  with  some  reser- 
vations as  to  former  grants,  and  was  the  remainder  of  the 
territory  conceded  to  the  United  States  under  the  defini- 
tive treaty  of  1783,  and  consisted  of  404,955.91  square 
miles,  or  259,171,787  acres.  This  was  the  public  domain 
of  the  United  States  on  April  30, 1803,  the  date  of  the 
Louisiana  purchase,  and  for  which 'the  original  survey 
and  disposition  laws  were  made. 

The  United  States  were  recognized  by  the  Crown  in  the 
definitive  treaty  of  peace  with  Great  Britain  as  "free 
sovereign  and  independent  States,  and  that  he  treats  with 
them  as  such,  and  for  himself,  his  heirs,  and  successors 
relinquishes  all  claims  to  the  government,  proprietary 
and  territorial  rights  of  the  same,  and  every  part  thereof." 

The  Government  of  the  United  States  acquired  as  cus- 
todian for  the  Nation,  lands  known  as  the  public  domain 
as  follows: 

From  States  (colonies  prior  to  July  4, 1776)  ceded  under 
the  Confederation  and  under  the  Constitution. 

This  was  in  pursuance  of  a  resolution  of  the  Congress 
of  the  Confederation  passed  Tuesday,  October  10,  1780, 
providing  for  the  reception  and  care  of  such  unappropri- 


188 


A   MANUAL  OP   LAND  SUBVEyiNG. 


ated  lands  as  might  be  ceded  by  States  to  the  United 
States,  and  for  the  disposition  of  the  same  for  the  com- 
mon benefit  of  the  United  States. 

The  dates  of  cession  of  these  lands  to  the  United  States 
were  as  follows: 


Colony. 

State. 

Date  of  Cession. 

New  Hampshire  

New  Hampshire 

No  cession 

New  York        .  ...        .. 

New  York 

March  1  1781 

Khode  Island  and  Provi- 
dence Plantations  

New  Jersey 

Rhode  Island. 

No  cession. 
Do 

New  Castle,  Kent  and  Sus- 
sex, on  Delaware  

Delaware 

Do 

Pennsylvania  

Do 

Virginia  

Virginia 

March  l    1734    and  De- 

Maryland 

Maryland. 

cember  30,  1788.* 
No  cession. 

Massachusetts  Bay  

Connecticut 

Massachusetts. 
Connecticut 

April  19,  1785. 
September  13  1786*  con- 

South Carolina  
North  Carolina  
Georgia  

South  Carolina. 
North  Carolina. 
Georgia. 

firmed  May  30,  1800.  - 
August  9,  1787. 
February  25,  1790. 
April  24,  1802. 

*An  act  to  change  the  conditions  of  the  cession  of  March  1, 1784,  only 
so  far  as  to  ratify  the  fifth  article  of  the  compact  of  the  ordinance  of 
1787. 

AREA  OF  CESSIONS. 


•  Sq.  miles. 

Acres. 

Massachusetts  (disputed)  claimed  (estimated)* 
Connecticu1  iuuim'tm)  and  Western  Keserve 
and  Fire-lands  (estimated)*  

54.000.00 
40.000.00 

34,560,000 
25,600.000 

From  New  York  and  Massachusetts  cession, 

315.91 

202,187 

From  Virginia  (disputed  and  undisputed)  to 
the  United  States  (exclusive  of  Kentucky  and 
including  area  of  Western  Eeserve  and  the 
Fire-lands)t  

265  562  00 

1  69  959  680 

South  Carolina  cession                                       .. 

4  900  00 

3  136  000 

North  Carolina  cession,  nominal,  because  the 
area  of  Tennessee  was  almost  covered  with 
reservations 

45  600  00 

29  184  000 

Georgia  cession             .. 

88  578  00 

56.689  920 

Total  actual  State  cessions  to  the  United 
States  for  public  domain 

404  955  91 

259  171  787 

*The  area  above  was  also  claimed  by  Virginia  and  included  in  her 
cession. 

tConnecticut's  jurisdictional  cession  of  the  Western  Reserve  and 
Fire-lands,  containing  about  3.800.000.  included  under  Virginia  cession. 


OKIGINAL   SURVEYS.  189 

AREA   OF  PURCHASES — PUBLIC   AND  NATIONAL   DOMAIN. 


Sq.  miles. 

Acres. 

Louisianaipurchase  April  30  1803  

1,182,752 

756  961,280 

East  and  West  Florida.  Feb.  22,  1819  
Gaudalupe  Hidalgo  February  2  1848 

59,268 
522  568 

37,931,520 
334  443  520 

State  of  Texas,  November  25,  1850  
Gadsden  purchase.  December  30,  1853...,  
Alaska  purchase,  March  30,  1867  

96,707 
45,535 
5/7,390 

61,892,480 
29,142,400 
369,529,600 

2,484,220;  1 ,589,900,800 


At  a  total  cost  of  §88,157,389.98. 

The  Texas  annexation  of  1845  added  to  the  national 
domain  the  area  of  the  present  State  of  Texas,  viz., 
274,356  square  miles,  or  175,587,840  acres,  included  in  the 
national  domain,  besides  the  purchase  of  1850  from  the 
State,  now  public  domain. 

The  total  area  of  purchased  and  annexed  territory,  in- 
cluded in  the  national  and  public  domain  since  1803,  is 
2, 758,576  square  miles,  or  1,765^468,640  acres.  This 
does  not  include  the  islands  acquired  from  Spain  in 
the  recent  Spanish  war. 

5.  The  present  system  of  survey  of  the  public 
lands  was  inaugurated  by  a  committee  appointed  by  the 
Continental  Congress,  and  consisting  of  the  following 
delegates:  Hon.  Thomas  Jefferson,  chairman,  Virginia; 
Hon.  Hugh  Williamson,  North  Carolina ;  Hon.  David 
Howell,  Rhode  Island ,  Hon.  Elbridge  Gerry,  Massachu- 
setts; Hon.  Jacob  Read,  South  Carolina. 

On  the  7th  of  May,  1784,  this  committee  reported  "An 
ordinance  for  ascertaining  the  mode  of  locating  and  dis- 
posing of  lands  in  the  western  territory,  and  for  other 
purposes  therein  mentioned."  This  ordinance  required 
the  public  lands  to  be  divided  into  "hundreds"  of  ten 
geographical  miles  square,  and  those  again  to  be  sub- 
divided into  lots  of  one  mile  square  each,  to  be  numbered 
from  1  to  100,  commencing  in  the  north-western  corner, 
and  continuing  from  west  to  east  and  from  east  to  west 


190  A  MANUAL  OF   LAND   SURVEYING. 

consecutively.  This  ordinance  was  considered,  debated, 
and  amended,  and  reported  to  Congress  April  26, 1785,  and 
required  the  surveyors  "  to  divide  the  said  territory  into 
townships  of  7  miles  square,  by  lines  running  due  north 
and  south,  and  others  crossing  these  at  right  angles.  *  *  * 
The  plats  of  the  townships,  respectively,  shall  be  marked 
by  subdivisions  into  sections  of  1  mile  square,  or  640 
acres,  in  the  same  direction  as  the  external  lines,  and 
numbered  from  1  to  49.  *  *  *  And  these  sections  shall  be 
subdivided  into  lots  of  320  acres." 

This  is  the  first  record  of  the  use  of  the  terms  "  town- 
ship "  and  "  section." 

May  3,  1785,  on  motion  of  Hon.  William  Grayson,  of 
Virginia,  seconded  by  Hon.  James  Monroe,  of  Virginia, 
the  section  respecting  the  extent  of  townships  was 
amended  by  striking  out  the  words  "seven  miles  square" 
and  substituting  the  words  "  six  miles  square."  The  rec- 
ords of  these  early  sessions  of  Congress  are  not  very  full 
or  complete;  but  it  does  not  seem  to  have  occurred  to  the 
members  until  the  6th*  of  May,  1785,  that  a  township  six 
miles  square  could  not  contain  49  sections  of  1  mile  square. 
At  that  date  a  motion  to  amend  was  made,  which  pro- 
vided, among  other  changes,  Lhat  a  township  should  con- 
tain 36  sections;  and  the  amendment  was  lost.  The  or- 
dinance as  finally  passed,  however,  on  the  20th  of  May, 
1785,  provided  for  townships  6  miles  square,  containing 
36  sections  of  1  mile  square.  The  first  pnblic  surveys 
were  made  under  this  ordinance.  The  townships,  6  miles 
square,  were  laid  out  in  ranges,  extending  northward 
from  the  Ohio  River,  the  townships  being  numbered  from 
south  to  north,  and  the  ranges  from  east  to  west.  The 
region  embraced  by  the  surveys  under  this  law  forms  a 
part  of  the  present  State  of  Ohio,  and  is  usually  styled 
"  The  Seven  Ranges."  In  these  initial  surveys  only  the 
exterior  lines  of  the  townships  wrere  surveyed,  but  the 
plats  were  marked  by  subdivisions  into  sections  of  1  mile 
square,  and  mile  corners  were  established  on  the  town- 


OBIGINAL   SURVEYS. 


191 


ship  lines.  The  sections  were  numbered  from  1  to  36, 
commencing  with  Ko.  1  in  the  southeast  corner  of  the 
township,  and  running  from  south  to  north  in  each  tier 
to  Xo.  36  in  the  northwest  corner  of  the  township,  as 
shown  in  the  following  diagram: 


36 

30 

24 

18 

12 

6 

35 

29 

23 

17 

—  £  , 
11 

6 

34 

28 

22 

16 

10 

4 

33 

27 

21 

15 

9 

3 

32 

26 

20 

14 

8 

2 

31 

25 

19 

13 

7 

'  1 

The  surveys  were  made  under  the  direction  of  the 
Geographer  of  the  United  States. 

The  act  of  Congress  approved  May  18, 1796,  provided 
for  the  appointment  of  a  surveyor-general,  and  directed 
the  survey  of  the  lands  northwest  of  the  Ohio  River, and 
above  the  mouth  of  the  Kentucky  River,  "  in  which  the 
titles  of  the  Indian  tribes  have  been  extinguished."  Un- 
der this  law  one-half  of  the  townships  surveyed  were 
subdivided  into  sections  "  by  running  through  the  same, 
each  way,  parallel  lines  at  the  end  of  every  two  miles, 
and  by  making  a  corner  on  each  of  said  lines  at  the  end 
of  every  mile,"  and  it  further  provided  that  "  the  sections 
shall  be  numbered,  respectively,  beginning  with  the  num- 
ber one  in  the  northeast  section  and  proceeding  west  and 
east  alternately,  through  the  township,  with  progressive 
numbers  till  the  thirty-sixth  be  completed."  This  method 


192 


A  MANUAL   Otf  LAND   SURVEYING. 


of  numbering  sections,  as  shown  by  the  following  dia- 
gram, is  still  in  use : 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

11 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

The  act  of  Congress  approved  May  10,  1800,  required 
the  "  townships  west  of  the  Muskingum,  which  *  *  *  are 
directed  to  be  sold  in  quarter  to \vnships,  to  be  subdivided 
into  half  sections  of  three  hundred  and  twenty  acres  each, 
as  nearly  as  may  be,  by  running  parallel  lines  through  the 
same  from  east  to  west,  and  from  south  to  north,  at  the 
distance  of  one  mile  from  each  other,  and  marking  cor- 
ners, at  the  distance  of  each  half  mile  on  the  lines  run- 
ning from  east  to  west,  and  at  the  distance  of  each  mile 
on  those  running  from  south  to  north.  *  *  *  And  the 
interior  lines  of  townships  intersected  by  the  Muskingum, 
and  of  all  the  townships  lying  east  of  that  river,  which 
have  not  been  heretofore  actually  subdivided  into  sec- 
tions, shall  also  be  run  and  marked.  *  *  *  And  in  all 
cases  where  the  exterior  lines  of  the  townships  thus  to  be 
subdivided  into  sections  or  half  sections  shall  exceed,  or 
shall  not  extend,  six  miles,  the  excess  or  deficiency  shall 
be  specially  noted,  and  added  to  or  deducted  from  the 
western  and  northern  ranges  of  sections  or  half  sections 
in  such  township,  according  as  the  error  may  be  in  run- 
ning the  lines  from  east  to  west  or  from  south  to  north." 


ORIGINAL   SURVEYS.  193 

6.  The  acts  of  Congress  defining  the  system  of  public 
land  surveys,  and  the  principles  to  be  employed  in  carry- 
ing them  out,  are  to  be  found  in  the  United  States  Stat- 
utes as  follows: 

Act  of  May     18, 1796,    Volume    1,  Chap.    29. 
10,1800,  "         2,      "        55. 

"     Feb.      11,1805,       "  ^       2,      "        li. 
"      April    24,1820,        **'•      3,      "        51. 
5,1832,       ,>'.      4,      "        55". 
.'«*     May       30,1862,  "        12,     "        86. 

"      March     3,1875,  "        18,      "      130. 

"      ;£*;  I      3,1875,        ,,_a.      JL9,    •-*      105. 
Such  portions  of  the  various  acts  as  are  now  in  force 
are  published  by  the  government  in  a  volume  entitled 
"  Existing  Land  Laws."    Those  Sections  which  refer  di- 
rectly to  the  surveys  are  as  follows: 

7.  United  States  Laws  relating-  to  Surveys  and 
Surveyors.— SEC.  77.  There  shall  be  appointed  by  the 
President,  by  and  with  the  advice  and  consent  of  the 
Senate,  a  surveyor-general  for  the  States  and  Territories 
herein  named,  embracing,  respectively,  one  surveying 
district,  namely:  Louisiana,  Florida,  Minnesota,  Kansas, 
California,  Xevada,  Oregon,  Nebraska  and  Iowa,  Dakota, 
Colorado,  New  Mexico,  Idaho,  Washington,    Montana, 
Utah,  Wyoming,  Arizona. 

3  Stat.  755;  4  id.  492;  9  id.  496;  10  id.  244,  306,  308,  309,  611;  11  id.  212; 
12  id.  176,  211,  214;  11  id.  77,  85.  314,  542;  15  id.  91 ;  16  id.  65,  240;  17 
id.  76;  18  id.  18,  34,  121,122,  123,  201,  303;  19  id.  126,  2075  R.  S.  2207. 

SEC.  84,  Every  surveyor-general  shall,  before  entering 
on  the  duties  of  his  office,  execute  and  deliver  to  the  Sec- 
retary of  the  Interior  a  bond,  with  good  and  sufficient 
security,  for  the  penal  sum  of  thirty  thousand  dollars, 
conditioned  for  the  faithful  disbursement,  according  to 
law,  of  all  public  money  placed  in  his  hands,  and  for  the 
faithful  performance  of  the  duties  of  his  office;  and  the 
President  has  discretionary  authority  to  require  a  new 

13 


194  A   MANUAL   OF   LAND   SURVEYING. 

bond  and  additional  security,  under  the  direction  of  the 
Secretary  of  the  Interior,  for  the  lawful  disbursement  of 
public  moneys. 

3  Stat.  697 ;  ft.  S.  2215,  2216,   U.  S.  v.  Vanzandt,  11  Wheat,  184;  U.  S.  v. 

Tingey,  6  Pet.  115;  Farrar  and  Brown  v.  U.  S,,  5  id.  373;  U.  S.  v. 

Bradley,  10  id.  343;  U.  S.  vs.  Linn,  15  id.  290;  U.  S.  v.  Prescott,3 

How.  578;  U.  8.  v.  Boyd,  5  id.  29;  Bryan  v.  U.  S.,  1  Black,  140;  Bov- 

den  v.  United  States,  13  Wall.  17;  Bevans  v.  U.  S  ,  13 id.  56;  U.  8. 

v.  Thomas,  15  id.  337;  U.S.  v.  Stephenson,  l  McClean,  C.  C.  462; 

U.  S.  v.  Linn,  2  id.  501 ;  U.  S.  v.  Ward,  3  id.  179.    8  Op.  Att.  Gen.  7. 

Cir.  G.  L.  O.,  July  1, 1871 ;  id.  May  14, 1879.    Treasury  Cir.,  July  13, 

1871  (Copp's  L.  L.  783;  1  Lester's  L.  L.  312,  314). 

SEC.  85.  The  commission  of  each  surveyor-general  shall 
cease  and  expire  in  four  years  from  the  date  thereof,  un- 
less sooner  vacated  by  death,  resignation,  or  removal 
from  office. 

3  Stat.  697;  E.  S.  2217.  Best  r.  Polk,  18  Wall.  112.  Decision  Com. 
G.  L.  O.,  Feb.  20, 1858  (1  Lester's  L.  L.  340). 

SEC.  8C.  Every  surveyor-general,  except  where  the  Pres- 
ident sees  cause  otherwise  to  determine,  is  authorized  to 
continue  in  the  uninterrupted  discharge  of  his  regular 
official  duties  after  the  day  of  expiration  of  his  commis- 
sion and  until  a  new  commission  is  issued  to  him  for  the 
same  office,  or  until  the  day  when  a  successor  enters  upon 
the  duties  of  such  office;  and  the  existing  official  bond  of 
any  officer  so  acting  shall  be  deemed  good  and  sufficient 
and  in  force  until  the  date  of  the  approval  of  a  new  bond 
to  be  given  by  him,  if  recom  missioned,  or  otherwise,  for 
the  additional  time  he  may  so  continue  officially  to  act, 
pursuant  to  the  authority  of  this  section. 

10  Stat.  247;  18  id,  62;  K.  S.  2222. 

SEC.  87.  Whenever  the  surveys  and  records  of  any  sur- 
veying distri  ct  are  com  pleted,  the  surveyor-general  thereof 
shall  be  required  to  deliver  over  to  the  Secretary  of  State 
of  the  respective  states,  including  such  surveys,  or  to 
such  other  officer  as  may  be  authorized  to  receive  them, 
all  the  field-notes,  maps,  records,  and  other  papers  apper- 
taining to  land  titles  within  the  same;  and  the  office  of 


ORIGINAL  SURVEYS.  195 

surveyor-general  in  every  such  district  shall  thereafter 
cease  and  be  discontinued. 

5  Stat.  384;  19  id.  121 ;  R.  S.  2218. 

SEC.  88.  In  all  cases  of  discontinuance,  as  provided  in 
the  preceding  section,  the  authority,  powers,  and  duties 
of  the  surveyor-general  in  relation  to  the  survey,  resur- 
vey,  or  subdivision  of  the  lands  therein,  and  all  matters 
and  things  connected  therewith,  shall  be  vested  in  and 
devolved  upon  the  Commissioner  of  the  General  Land 
Office. 

10  Stat.  152;  R.S.2219. 

SEC.  89.  Under  the  authority  and  direction  of  the  Com- 
missioner of  the  General  Land  Office,  any  deputy  surveyor 
or  other  agent  of  the  United  States  shall  have  free  access 
to  any  such  field-notes,  maps,  records,  and  other  papers 
for  the  purpose  of  taking  extracts  therefrom  or  making 
copies  thereof  without  charge  of  any  kind;  but  no  transfer 
of  such  public  records  shall  be  made  to  the  authorities  of 
any  State  until  such  State  has  provided  by  law  for  the 
reception  and  safe-keeping  of  such  public  records  and  for 
the  allowance  of  free  access  thereto  by  the  authorities  of 
the  United  States. 

10  Stat.  152;  18  id.  62;  R.  S.  2220,  2221. 

SEC.  90.  Every  surveyor-general  shall  engage  a  sufficient 
number  of  skillful  surveyors  as  his  deputies,  to  whom 
he  is  authorized  to  administer  the  necessary  oaths  upon 
their  appointments.  He  shall  have  authority  to  frame 
regulations  for  their  direction,  not  inconsistent  with  law 
or  the  instructions  of  the  General  Land  Office,  and  'to 
remove  them  for  negligence  or  misconduct  in  office. 

Taylor  and  Quarlls  v.  Brown,  5  Cranch,  234;  Craig  et  al.  v.  Braxford, 
3  Wheat,  594;  Ellicott  et  al.  v.  Pearl,  10  Pet.  412;  Brown's  Lessee 
v.  Clements,  3  How.  650.  Reed  v.  Con  way  20  Mo.  22;  same  case, 
26  id,  13;  Hamil  v.  Carr,  21  Ohio  St.  258;  Doe  v.  Hildreth,  2  Irid. 
274;  McClintock  v.  Rodgers,  11  Ills.  279.  Cir.  G.  L.  O.,  June  26, 
1880. 

Second.  He  shall  cause  to  be  surveyed,  measured,  and 
marked,  without  delay,  all  base  and  meridian  lines  through 


196  A  MANUAL  OP   LAND  SURVEYING. 

such  points  and  perpetuated  by  such  monuments,  and 
such  other  correction  parallels  and  meridians  as  may  be 
prescribed  by  law  or  by  instructions  from  the  General 
Land  Office  in  respect  to  the  public  lands  within  his  sur- 
veying district,  to  which  the  Indian  title  has  been  or  may 
be  hereafter  extinguished. 

Gazzen  v.  Phillips'  Lessee,  20  How.  372.  3  Op.  Att.  Gen.,  281,  284. 
Atshire  v.  Hulse,  1  Ohio,  170;  Hastings  v.  Stevenson,  2  d.  9;  Mc- 
Kinney  v.  McKinney,  8  id.  423;  Eamil  v.  Carr,  21  Ohio  St.  258; 
Hendrick  v.  Eno,  42  Iowa  411 ;  Saint  Louis  v.  Walker,  40  Mo.  383; 
Jordan  v.  Barrett,  13  La.  24;  Fowler  i>.  Duval,  11  id.  5C1;  Cox  v. 
Jones,  47  Cal.  412.  Cir.  G.  L.  O.,  June  26, 1880. 

Third.  He  shall  cause  to  be  surveyed  all  private  land 
claims  within  his  district  after  they  have  been  confirmed 
by  authority  of  Congress,  so  far  as  may  be  necessary  to 
complete  the  survey  of  the  public  lands. 

Menard's  Heirs  v.  Massey,  8  How.  293;  Kissell  v.  St.  Louis  Public 
Schools,  18  id.  19;  Stanford  v.  Taylor,  18  id.  409;  Ballance  v.  For- 
syth,  24  id.  183;  U.  S.  v.  Fossat,  25  id.  445;  Carondelet  v.  St.  Louis, 
1  Black,  179;  U.  S.  v.  Sepulveda,  1  Wall.  104;  U.  S.  v.  Halleck,  1  id. 
439;  U.  S.  v.  Billings,  2  id.  444;  Sutler's  case,  2  id.  562;  U.  S.  v. 
Pacheco,  2  id.  587;  Fossat  case,  2  id.  649;  Dehon  v.  Bernal,  2  id. 
774;  U.  S.  v.  Armijo,  5  Cd.  444;  Higueras  v.  U.  S.  5  id.  827;  Maguire 
v.  Tyler,  8  id.  650;  Lynch  v.  Bernal  9  id.  315;  Henshaw  v.  Bissell, 
18  id.  255;  Shepley  et  al.  v.  Cowan  et  al.,  1  Otto,  330;  Miller  et  al.  vt 
Dale  et  al,  2  id.  473;  Van  Eeynegand  v.  Bolton,  6  id.  33;  U.  S.  v'. 
Throckmorton,  8  id.  61 ;  Snyder  v.  Sickles,  8  id.  203;  Scull  v.  U.  S., 
8  id,  410.  Bissell  v.  Henshaw,  1  Saw.  C.  C.  553;  Leroy  v.  Jamison, 
3  id.  369.  Gibson  v.  Chouteau,  39  Mo.  536 ;  Milburn  v.  Hardy,  28  id. 
514;  Funkhouser  v.  Hantz,  29  id.  540;  Dent  v.  Legesson,  29  id.  489; 
Carondelet  v.  St.  Louis,  29  id.  527;  Maguire  v.  Tyler,  30  id.  202; 
Robins  v.  Eckler,  36  id.  494;  Clark  v.  Heammerle,  36  id.  620;  Gib- 
son v.  Chouteau,  39  id.  536;  Vasquez  v.  Ewing,  42  id.  247;  Glasgow 
v.  Lindell,50id.  60;  Eector  v.  Gaines,  19  Ark.  70 ;  Ashley  v.Kector, 
20  id.  359;  Meaux  v.  Breaux,  10  Martin  (La.)  364;  Moon  v.  Wilkin- 
son, 13  Cal.  478;  Boggs  v.  Mining  Co.,  14  id.  279;  Mott  v.  Smith,  16 
id.  534 ;  Johnson  v.  Van  Dyke,  20  id.  225 ;  McGarrahan  v.  Maxwell, 
27  id.  75;  Treadway  v.  Scmple,  28  id.  652;  Searle  v.  Ford,  29  id.  104; 
Mahoney  v.  Van  Winkle,  33  id.  448;  Morrill  v.  Chapman,  35  id.  85; 
Yates  v.  Smith,  38  id.  60;  San  Diego  u.  Allison,  46  id.  163.  De- 
cisions Sec.  Int.,  July  1C,  1872;  Aug.  8, 1876;  Aug.  17,  1876;  March 
16,1877.  Decisions  Com.  G.  L.  O.,  Aug.  18,  1860;  Sept.  18,  1874; 
Nov.  3, 1874;  Sept.  18, 1875;  Oct.  28, 1875;  June  26, 1879.  Cir.  G.  L. 
O.,  June  26, 1880. 


ORIGINAL  SURVEYS.  197 

Fourth.  He  shall  transmit  to  the  register  of  the  respec- 
tive  land  offices  within  his  district  general  and  particular 
plats  of  all  lands  surveyed  by  him  for  each  land  district; 
and  he  shall  forward  copies  of  such  plats  to  the  Commis- 
sioner of  the  General  Land  Office. 

Barnard  v.  Ashley,  18  How.  43;  Water  and  Mining  Co.  v .  Bugbee,  6 
Otto.  1G5;  Hamil  v.  Carr,  21  Ohio  St.  258;  Doe  v.  Hildreth,  2  Ind 
274;  Pope  v.  Athearn,  42  Cal.  606;  Com.  G.  L.  O.  Instructions  to 
Surveyor-General,  April  17, 1879. 

Fifth.  He  shall,  so  far  as  is  compatible  with  the  desk 
duties  of  his  office,  occasionally  inspect  the  surveying 
operations  while  in  progress  in  the  field,  sufficiently  to 
satisfy  himself  of  the  fidelity  of  the  execution  of  the 
work  according  to  contract;and  the  actual  and  necessary 
expenses  incurred  by  him  while  so  engaged  shall  be 
allowed ;  and  where  it  is  incompatible  with  his  other  duties 
for  a  surveyor-general  to  devote  the  time  necessary  to 
make  a  personal  inspection  of  the  work  in  progress,  then 
he  is  authorized  to  depute  a  confidential  agent  to  make 
such  examination,  and  the  actual  and  necessary  expenses 
of  such  person  shall  be  allowed  and  paid  for  that  service, 
and  five  dollars  a  day  during  the  examination  in  the  field; 
but  such  examination  shall  not  be  protracted  beyond 
thirty  days,  and  in  no  case  longer  than  is  actually  neces- 
sary; and  when  a  surveyor-general,  or  any  person  em- 
ployed in  his  office  at  a  regular  salary,  is  engaged  in  such 
special  service  he  shall  receive  only  his  necessary  expenses 
in  addition  to  his  regular  salary. 

1  Stat.  464;  13 id.  325;  4  id.  492;  10  id.  245,  247;  18  id.  34;  19  id.  126;  R. 
8.2223.  Sec.  Int.  Instructions,  July  l,  1874;  Sept.  21,  1874.  Cir. 
G.  L.  O.,  June  26, 1880. 

SEC.  91.  Every  deputy  surveyor  shall  enter  into  a  bond, 
with  sufficient  security,  for  the  faithful  performance  of 
all  surveying  contracts  confided  to  him:  and  the  penalty 
of  the  bond,  in  each  case,  shall  be  double  the  estimated 
amount  of  money  accruing  under  such  contracts,  at  the 
rate  per  mile  stipulated  to  be  paid  therein.  The  suffici- 


198  A  MANUAL  OF  LAND  SURVEYING. 

ency  of  the  sureties  to  all  such  bonds  shall  be  approved 
and  certified  by  the  proper  surveyor-general. 

4  Stat.  493;  10  id.  247;  R.  S.  2230.    U.  S.  v.  Vanzandt,  11  Wheat.  184; 

U.  S.  v.  Tingey,  5  Pet.  115;  Farrar  et  al.  v.  U.  S.,  5  id.  373;  U.  S.  v. 

Bradley,  10  id.  343;  U.  S.  v.  Linn,  15  id.  290.    U.  S.  v.  Stephenson, 

1  McLean,  C  C.  462. 

SEC.  92.  The  surveyors-general,  in  addition  to  the  oath 
now  authorized  by  law  to  be  Administered  to  deputies  on 
their  appointment  to  office,  shall  require  each  of  their 
deputies,  on  the  return  of  his  surveys,  to  take  and  sub- 
scribe an  oath  that  those  surveys  have  been  faithfully 
and  correctly  executed  according  to  law  and  the  instruc- 
tions of  the  surveyor-general. 

9  Stat.  79;  R.  S.  2231.  Ellicott  and  Meredith  v.  Pearle,  10  Pet.  412; 
U.  S.  v.  Hanson,  16  id.  196;  Bollard  et  al.  v.  Dwight  et  al.,  4  Cranch, 
421 ;  Taylor  et  al  v.  Brown,  5  id.  234.  Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  93.  The  district  attorney  of  the  United  States,  in 
whose  district  any  false,  erroneous,  or  fraudulent  surveys 
have  been  executed,  shall,  upon  the  application  of  the 
proper  surveyor-general,  immediately  institute  suit  upon 
the  bond  of  such  deputy,  and  the  institution  of  such  suit 
shall  act  as  a  lien  upon  any  property  owned  or  held  by 
such  deputy  or  his  sureties  at  the  time  such  suit  was 
instituted. 

9  Stat.  79;  R.S.2232. 

SEC.  99.  The  public  lands  shall  be  divided  by  north  and 
south  lines  run  according  to  the  true,  meridian,  and  by 
others  crossing  them  at  right  angles,  so  as  to  form  town- 
ships of  six  miles  square,  unless  where  the  line  of  an 
Indian  reservation,  or  of  tracts  of  land  heretofore  sur- 
veyed or  patented,  or  the  course  of  navigable  rivers,  may 
render  this  impracticable;  and  in  that  case  this  rule  must 
be  departed  from  no  further  than  such  particular  circum- 
stances require. 

McKinney  v,  McKinney,  8  Ohio,  423;  Hamil  v.  Carr,  21  Ohio  St.  258. 
Decision  Sec.  Int ,  Jan.  24, 1880.  Cir.  G.  L.  O  ,  June  26, 1880. 

Second.  The  corners  of  the  townships  must  be  marked 
with  progressive  numbers  from  the  beginning,  each  dis- 


ORIGINAL  SURVEYS.  199 

tance  of  a  mile  between  such  corners  must  be  also  dis- 
tinctly marked  with  marks  different  from  those  of  the 
corners. 

Third.  The  township  shall  be  subdivided  into  sections, 
containing,  as  nearly  as  may  be,  six  hundred  and  forty 
acres  each,  by  running  through  the  same,  each  way,  par- 
allel lines  at  the  end  of  every  two  miles;  and  by  making 
a  corner  on  each  of  such  lines,  at  the  end  of  every  mile. 
The  sections  shall  be  numbered,  respectively,  beginning 
with  the  number  one  in  the  northeast  section  and  pro- 
ceeding west  and  east  alternately  through  the  township 
with  progressive  numbers  till  the  thirty-six  be  completed. 

Grogan  r.  Knight,  27  Ccl.  516.  Decision  Sec.  Int.,  April  14, 1879.  Cir. 
G.  L.  O.,  June  26, 1880. 

Fourth.  The  deputy  surveyors,  respectively,  shall  cause 
to  be  marked  on  a  tree  near  each  corner  established  in  the 
manner  described,  and  within  the  section,  the  number  of 
such  section,  and  over  it  the  number  of  the  township 
within  which  such  section  may  be;  and  the  deputy  sur- 
veyors shall  carefully  note,  in  their  respective  field-books, 
the  names  of  the  corner-trees  marked  and  the  numbers 
so  made. 

Cir.  G.  L.  O.,  June  26, 1880. 

Fifth.  Where  the  exterior  lines  of  the  townships  which 
may  be  subdivided  into  sections  or  half -sections  exceed, 
or  do  not  extend  six  miles,  the  excess  or  deficiency  shall 
be  specially  noted,  and  added  to  or  deducted  from  the 
western  and  northern  ranges  of  sections  or  half-sections 
in  such  townships,  according  as  the  error  may  be  in  run- 
ning the  lines  from  east  to  west,  or  from  north  to  south; 
the  sections  and  half-sections  bounded  on  the  northern 
and  western  lines  of  such  townships  shall  be  sold  as  con- 
taining only  the  quantity  expressed  in  the  returns  and 
plats  respectively,  and  all  others  as  containing  the  com- 
plete legal  quantity. 

Knight  v.  Elliott,  57  Mo.  317;  Vaughn  v.  Tate,  64  id.  491;  Walters  v. 
Commons,  2  Port.  (Ala-)  38;  Lewen  r.  Smith,  7  id.  428.  Decision 
Sec.  Int.,  April  14, 1879,  Cir.  G.  L.  O.,  June  26, 1880. 


200.  A  MANUAL  OF  LAND  SURVEYING. 

Sixth.  All  lines  shall  be  plainly  marked  upon  trees,  and 

measured  with  chains,  containing  two  perches  of  sixteen 

and  one-half  feet  each,  subdivided  into  twenty-five  equal 

links;  and  the  chain  shall  be  adjusted  to  a  standard  to  be 

'  kept  for  that  purpose. 

Bradley  v.  Taylor,  5  Crancli,  191 ;  Mclvers  v.  Walker,  9  id.  173;  Shipp 
v.  Miller's  Heirs,  2  Wheat.  316;  Holmes  v.  Trout,  7  Pet.  171;  Brown 
v.  Huger,  21  How.  303;  Meron  v.  Whitney,  5  Otto,  551;  Robinson 
v.  Moon,  4  McLean,  C.  C.  279.  Oakley  v.  Stuart,  52  Cal.  521.  Cir. 
G.  L.  O.,  June  26,  1880. 

Seventh.  Every  surveyor  shall  note  in  his  field-book  the 
true  situations  of  all  mines,  salt  licks,  salt  springs,  and 
and  mill-seats  which  come  to  his  knowledge;  all  water 
courses  over  which  the  line  he  runs  may  pass;  and  also 
the  quality  of  the  lands. 

Newsom  v.  Pryor's  Lessee,  7  Wheat.  7;  Preston  v.  Bowman,  6  id.  580; 
Patterson  v  Jenks,  2  Pet.  216. 

Eighth.  These  field  books  shall  be  returned  to  the  sur- 
veyor-general, who  shall  cause  therefrom  a  description  of 
the  whole  lands  surveyed  to  be  made  out  and  transmitted 
to  the  officers  who  may  superintend  the  sales.  He  shall 
also  cause  a  fair  plat  to  be  made  of  the  townships  and 
fractional  parts  of  townships  contained  in  the  lands,  de- 
scribing the  subdivisions  thereof  and  the  marks  of  the 
corners.  This  plat  shall  be  recorded  in  books  to  be  kept 
for  that  purpose;  and  a  copy  thereof  shall  be  kept  open 
at  the  surveyor-general's  office  for  public  information, 
and  other  copies  shall  be  sent  to  the  places  of  the  sale 
and  to  tne  General  Land  Office. 

1  Stat.  465;  2  id.  73;  19  id.  348;  K.  S.  2395.  Taylor  et  al.  v.  Brown,  5 
Crancli,  234;  Barnard  v.  Ashley,  18  How.  43;  Water  and  Mining 
Co.  v.  Bugbee,  6  Otto,  165.  Eector  v.  Gaines,  19  Ark.  70;  Lewen  v, 
Smith,  5  Port.  (Ala.)  428 ;  Mptt  v.  Smith,  16  Cal.  534;  Hamil  v.  Carr, 
21  Ohio  St.  258;  Doe  v.  Hildreth,  2  Ind.  274;  McClintock  v.  Eod- 
gers,  11  Ills.  279.  Decision  Sec.  Int.,  Jan.  15, 1878  Decision  Com. 
G.  L.  O.,  April  17,  1879. 

SEC.  100.  The  boundaries  and  contents  of  the  several 
sections,  half-sections,  and  quarter-sections  of  the  public 


ORIGINAL  SURVEYS.  201 

lands  shall  be  ascertained  in  conformity  with  the  follow- 
ing principles: 

First.  All  the  comers  marked  in  the  surveys,  returned 
by  the  surveyor-general,  shall  be  established  as  the  proper 
corners  of  sections,  or  subdivisions  of  sections,  which 
they  were  intended  to  designate;  and  the  corners  of  half 
and  quarter  sections,  not  marked  on  the  surveys,  shall  be 
placed  as  nearly  as  possible  equidistant  from  those  two 
corners  which  stand  on  the  same  line. 

Second.  The  boundary  lines,  actually  run  and  marked 
in  the  surveys  returned  by  the  surveyor-general,  shall  be 
established  as  the  proper  boundary  lines  of  the  sections, 
or  subdivisions,  for  which  they  were  intended,  and  the 
length  of  such  lines,  as  returned,  shall  be  held  and  con- 
sidered as  the  true  length  thereof.  And  the  boundary 
lines  which  have  not  been  actually  run  and  marked  shall 
be  ascertained  by  running  straight  lines  from  the  estab- 
lished corners  to  the  opposite  corresponding  corners;  but 
in  those  portions  of  the  fractional  townships  where  no 
such  opposite  corresponding  corners  have  been  or  can  be 
fixed,  the  boundary  lines  shall  be  ascertained  by  running 
from  the  established  corners  due  north  and  south  or  east 
and  west  lines,  as  the  case  may  be,  to  the  water-course, 
Indian  boundary  line,  or  other  external  boundary  of  such 
fractional  township. 

Mott  v.  Smith,  1C  Cal.  534;  Guin  v.  Brandon.  2y  Ohio  St.  656;  McCIin- 
tock  r.  Rodgers,  11  Ills.  279;  Goodman  r,  Myriek,  5  Oreg.  65.  Cir. 
G.  L.  O.,  June  26, 1880. 

Third.  Each  section  or  subdivision  of  section,  the  con- 
tents whereof  have  been  returned  by  the  surveyor-gen- 
eral, shall  be  held  and  considered  as  containing  the  exact 
quantity  expressed  in  such  return ;  and  the  half-sections 
and  quarter-sections,  the  contents  whereof  shall  not  have 
been  thus  returned,  shall  be  held  and  considered  as  con- 
taining the  one-half  or  the  one-fourth  part,  respectively, 


202  A    MANUAL    OF    LAND    SURVEYING. 

of  the  returned  contents  of  the*  section  of  which  they 
make  part. 

2  Stat.  313;  R.  S.  2396.    Lindsey  v.  Hawes,  2  Black,  554;  U.  S.  v.  Pa- 

checo,  2  Wall.  587;  Kailway  Co.  v.  Schurmier,  7  id.  272;  County  of 
Saint  Clair  v.  Livingston,  23  id.  46;  Heidekoper  v.  Brooms,  1 
Wash.C.  C.  109;  Coon  v.  Ten,  1  Pet.  C.  C.  496.  2  Op.  Att.  Gen. 
578.  Knight  v.  Elliott,  57  Mo.  317;  Vaughn  v.  Tate,  64  id.  <91; 
Waters  v.  Commons,  2  Port.  (Ala.)  38 ;  Lewen  v.  Smith,  7  id.  428; 
Billingsly  v.  Bates,  30  Ala.  376 ;  Doe  r.  Hildreth,  2  Ind.  274;  Gro- 
gan  v.  Knight,  27  Cal.  516.  Decision  Com.  G.  L.  O.,  May  17, 1875. 
Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  101.  In  every  case  of  the  division  of  a  quarter-sec- 
tion the  line  for  the  division  thereof  shall  run  north  and 
south,  and  the  corners  and  contents  of  half  quarter-sec- 
tions which  may  thereafter  be  sold  shall  be  ascertained 
in  the  manner  and  on  the  principles  directed  and  pre- 
scribed by  the  section  preceding,  and  fractional  sections 
containing  one  hundred  and  sixty  acres  or  upwards  shall 
in  like  manner,  as  nearly  as  practicable,  be  subdivided 
into  half  quarter-sections,  under  such  rules  and  regula- 
tions as  may  be  prescribed  by  the  Secretary  of  the  Inte- 
rior, and  in  every  case  of  a  division  of  a  half  quarter- 
section,  the  line  for  the  division  thereof  shall  run  east 
and  west,  and  the  corners  and  contents  of  quarter  quarter- 
section,  which  may  thereafter  be  sold,  shall  be  ascertained, 
as  nearly  as  may  be,  in  the  manner  and  on  the  principles 
directed  and  prescribed  by  the  section  preceding;  and 
fractional  sections  containing  fewer  or  more  than  one 
hundred  and  sixty  acres  shall  in  like  manner,  as  nearly  as 
may  bo  practicable,  be  subdivided  into  quarter  quarter- 
sections,  under  such  rules  and  regulations  as  may  be  pre- 
scribed by  the  Secretary  of  the  Interior. 

3  Stat.  566  ;  4  id.  503 ;  R.  S.  2397.    Gazzam  r.  Phillips'  Lessee,  20  How. 

372 ;  Railway  Co.  v.  Schurmier,  7  Wall.  272.  Buel  v.  Tuley,  4  Mc- 
Lean, C.  C.  268.  Wharton  v.  Littlefield,  30  Ala.  245.  3  Op.  Att. 
Gen.  281,284.  Decision  Sec.  Int.,  April  14,1879.  Decision  Com. 
G.  L,  O.,  May  17, 1875.  Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  102.  Whenever,  in  the  opinion  of  the  President,  a 
departure  from  the  ordinary  method  of  surveying  land 


ORIGINAL    SURVEYS.  203 

on  any  river,  lake,  bayou,  or  water-.course  would  promote 
the  public  interest,  he  may  direct  the  surveyor-general, 
in  whose  district  such  land  is  situated,  and  where  the 
change  is  intended  to  be  made,  to  cause  the  lands  thus 
situated  to  be  surveyed  in  tracts  of  two  acres  in  width^ 
fronting  oa  any  river,  bayou,  lake,  or  water-course,  and 
running  back  the  depth  of  forty  acres;  which  tracts  of 
land  so  surveyed  shall  be  offered  for  sale  entire,  instead 
of  in  half  quarter-sections,  and  in  the  usual  manner,  and 
on  the  same  terms  in  all  respects  as  the  other  public  lands 
of  the  United  States. 

4  Stat.  34 ;  R.  S.  2407. 

• 
SEC.  103.  In  extending  the  surveys  of  the  public  lands 

in  the  State  of  Nevada,  the  Secretary  of  the  Interior  may 
vary  the  lines  of  the  subdivisions  from  a  rectangular 
form,  to  suit  the  circumstances  of  the  country. 

14  Stat.  86 ;  R,  S.  2408.    Heydenfeldt  v.  Mining  Co.,  3  Otto,  634. 

SEC.  104.  The  Secretary  of  the  Interior,  if  he  deems  it 
advisable,  is  authorized  to  continue  the  surveys  in  Ore- 
gon and  California,  to  be  made  after  what  is  known  as 
the  geodetic  method,  under  such  regulations  and  upon 
such  terms  as  have  been  or  may  hereafter  be  prescribed 
by  the  Commissioner  of  the  General  Land  Office;  but 
none  other  than  township  lines  shall  be  run  where  the 
land  is  unfit  for  cultivation;  nor  shall  any  deputy  sur- 
veyor charge  for  any  line  except  such  as  may  be  actually 
run  and  marked  or  for  any  line  not  necessary  to  be  run. 

9  Stat.  496 ;  10  id.  245 ;  K.  S.  2409. 

SEC.  105.  "Whenever,  in  the  opinion  of  the  Secretary  of 
the  Interior,  a  departure  from  the  rectangular  mode  of 
surveying  and  subdividing  the  public  lands  in  California 
would  promote  the  public  interests,  he  may  direct  such 
change  to  be  made  in  the  mode  of  surveying  and  desig- 
nating such  lands  as  he  deems  proper,  with  reference  to 
the  existence  of  mountains,  mineral  deposits,  and  the  ad- 
vantages derived  from  timber  and  water  privileges;  but 
such  lands  shall  not  be  surveyed  into  less  than  one  hun- 


204  A    MANUAL    OF    LAND    SURVEYING. 

dred  and  sixty  acres  or  subdivided  into  less  than  forty 
acres. 

10  Stat.  245 :  B.  S.  2410.    Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  106.  The  public  surveys  shall  extend  over  all  min- 
eral lands,  and  all  subdividing  of  surveyed  lands  into  lots 
less  than  one  hundred  and  sixty  acres  may  be  done  by 
county  and  local  surveyors  at  the  expense  of  claimants; 
but  nothing  contained  in  this  section  shall  require  the 
survey  of  waste  or  useless  lands. 

10  Stat.  15,  21 ;  16  id.  218 ;  E.  S.  2406. 

SEC.  107.  The  printed  manual  of  instructions  relating 
to  tne  puolic  survey*,  prepared  at  the  General  Land  Office, 
and  bearing  date  January  first,  nineteen  hundred 
and  three,  the  instructions  of  the  Commissioner  of 
the  General  Land  Office,  and  the  special  instructions 
of  the  surveyor-general,  when  not  in  conflict  with  such 
printed  manual  or  the  instructions  of  the  Commissioner, 
shall  be  taken  and  deemed  to  be  a  part  of  every  contract 
for  surveying  the  public  lands. 

12  Stat.  409 ;  K.  S.  2399.    Cir.  G.  L.  O.,  June  26,  1880. 

SEC.  108.  Legal  subdivisions  of  forty  acres  of  placer 
lands  may  be  subdivided  into  ten-acre  lots. 

16  Stat.  213 ;  R.  S.  2330. 

SEC.  2320.  Mining- claims  upon  veins  or  lodes  of  quartz 
or  other  rock  in  place  bearing  gold,  silver,  cinnabar,  lead, 
tin,  copper,  or  other  valuable  deposits,  heretofore  located, 
shall  be  governed  as  to  length  along  the  vein  or  lode  by 
the  customs,  regulations,  and  laws  in  force  at  the  date  of 
their  location.  A  mining-claim  located  after  the  tenth 
day  of  May,  eighteen  hundred  and  seventy-two,  whether 
located  by  one  or  more  persons,  may  equal,  but  shall  not 
exceed,  one  thousand  five  hundred  feet  in  length  along 
the  vein  or  lode;  but  no  location  of  a  mining-claim  shall 
be  made  until  the  discovery  of  the  vein  or  lode  within 
the  limits  of  the  claim  located.  No  claim  shall  extend 
more  than  three  hundred  feet  on  each  side  of  the  middle 


ORIGINAL    SURVEYS.  205 

of  the  vein  at  the  surface,  nor  shall  any  claim  be  limited 
by  any  mining  regulation  to  less  than  twenty-five  feet  on 
each  side  of  the  middle  of  the  vein  at  the  surface,  except 
where  adverse  rights  existing  on  the  tenth  day  of  May, 
eighteen  hundred  and  seventy-two,  render  such  limita- 
tion necessary.  The  end-lines  of  each  claim  shull  be 
parallel  to  each  other. 

10  May,  1872,  c.  152,  S.  2.  V.  17,  p.  91. 

SEC.  2322.  The  locators  of  all  mining  locations  hereto- 
fore made  or  which  shall  hereafter  be  made,  on  any  min- 
eral vein,  lode,  or  ledge,  situated  on  the  public  domain, 
their  heirs  and  assigns,  where  no  adverse  claim  exists  on 
the  tenth  day  of  May,  eighteen  hundred  and  seventy-two, 
so  long  as  they  comply  with  the  laws  of  the  United 
States,  and  with  State,  Territorial  and  local  regulations 
not  in  conflict  with  the  laws  of  the  United  States  govern- 
ing their  possessory  title,  shall  have  the  exclusive  right 
of  possession  and  enjoyment  of  all  the  surface  included 
within  the  lines  of  their  locations,  and  of  all  veins,  lodes, 
and  ledges  throughout  their  entire  depth,  the  top  or  apex 
of  which  lies  inside  of  such  surface-lines  extended  down- 
ward vertically,  although  such  veins,  lodes,  or  ledges  may 
so  far  depart  from  a  perpendicular  in  their  course  down- 
ward as  to  extend  outside  the  vertical  side-lines  of  such 
surface  locations.  But  their  right  of  possession  to  such 
outside  parts  of  such  veins  or  ledges  shall  be  confined  to 
such  portions  thereof  as  lie  between  vertical  planes 
drawn  downward  as  above  described,  through  the  end- 
lines  of  their  locations,  so  continued  in  their  own  direc- 
tion that  such  planes  will  intersect  such  exterior  parts  of 
such  veins  or  ledges.  And  nothing  in  this  section  shall 
authorize  the  locator  or  possessor  of  a  vein  or  lode  which 
extends  in  its  downward  course  beyond  the  vertical  lines 
of  his  claim  to  enter  upon  the  surface  of  a  claim  owned 
or  possessed  by  another. 

10  May,  1872,  C.  152,  S.  3,  V.  17,  p.  91. 

SEC.  2323.  Where  a  tunnel  is  run  for  the  development 
of  a  vein  or  lode,  or  for  the  discovery  of  mines,  the  own- 


206  A    MANUAL    OF    LAND    SURVEYING. 

ers  of  such  tunnel  shall  have  the  right  of  possession  of 
all  veins  or  lodes  within  three  thousand  feet  from  the 
face  of  such  tunnel  on  the  line  thereof,  not  previously 
known  to  exist,  discovered  in  such  tunnel,  to  the  same 
extent  as  if  discovered  from  the  surface;  and  locations 
on  the  line  of  such  tunnel  of  veins  or  lodes  not  appearing 
on  the  surface,  made  by  other  parties  after  the  commence- 
ment of  the  tunnel,  and  while  the  same  is  being  prose- 
cuted with  reasonable  diligence,  shall  be  invalid ;  but 
failure  to  prosecute  the  work  on  the  tunnel  for  six 
months  shall  be  considered  as  an  abandonment  of  the 
right  to  all  undiscovered  veins  on  the  line  of  such  tunnel. 

10  May,  1872,  C.  152,  S.  4,  V.  17,  p.  92. 

SEC.  2324.  The  miners  of  each  mining-district  may 
make  regulations  not  in  conflict  with  the  laws  of  the 
United  States,  or  with  the  laws  of  the  State  or  Territory 
in  which  the  district  is  situated,  governing  the  location, 
manner  of  recording,  amount  of  work  necessary  to  hold 
possession  of  a  mining-claim,  subject  to  the  following 
requirements:  The  location  must  be  distinctly  marked  on 
the  ground  so  that  its  boundaries  can  be  readily  traced. 
All  records  of  mining-claims  hereafter  made  shall  con- 
tain the  name  or  names  of  the  locators,  the  date  of  the 
location,  and  such  a  description  of  the  claim  or  claims 
located  by  reference  to  some  natural  object  or  permanent 
monument  as  will  identify  the  claim. 

10  May,  1872,  C.  152,  S.  5,  V.  17,  p.  92. 

SEC.  109.  The  surveyor-general  of  the  United  States 
may  appoint  in  each  land  district  containing  mineral 
lands  as  many  competent  surveyors  as  shall  apply  for  ap- 
pointment to  survey  mining  claims.  The  expenses  of  the 
survey  of  vein  or  lode  claims,  and  the  survey  and  sub- 
division of  placer  claims  into  smaller  quantities  than  one 
hundred  and  sixty  acres,  shall  be  paid  by  the  applicants, 
and  they  shall  be  at  liberty  to  obtain  the  same  at  the 
most  reasonable  rates,  and  they  shall  also  be  at  liberty  to 
employ  any  United  States  deputy  surveyor  to  make  the 


ORIGINAL    SURVEYS.  207 

survey.  The  Commissioner  of  the  General  Lund  Office 
shall  have  power  to  establish  the  maximum  charges  for 
such  surveys;  and  to  the  end  that  he  may  be  fully  in- 
formed on  the  subject,  each  applicant  shall  file  with  the 
register  a  sworn  statem3nt  of  all  charges  and  fees  paid 
by  such  applicant  for  surveys,  which  statement  shall  be 
transmitted  to  the  Commissioner  of  the  General  Land 
Office. 

17Stat.95;  19  id.  52;  R.  S.  2334.  Decision  Cora.  G.  L.  O.,  April  20, 
1877. 

SEC.  110.  The  surveyor-general  of  the  United  States 
shall  prepare  or  cause  to  be  prepared  a  plat  and  field-notes 
of  all  mining  surveys  made  by  authority  of  law,  which 
shall  show  accurately  the  boundaries  of  such  claims;  and, 
when  warranted  by  the  facts,  he  shall  give  to  the  claim- 
ant his  certificate  that  five  hundred  dollars*  worth  of 
labor  has  been  expended  or  improvements  made  upon  the 
claim  by  the  claimant  or  his  grantors,  and  that  the  plat 
is  correct,  with  such  further  description  by  such  refer- 
ence to  natural  objects  or  permanent  monuments  as  shall 
identify  the  claim,  and  furnish  an  accurate  description. 
to  be  incorporated  in  the  patent. 

17Stat.92    R.  S.2325 

SEC.  111.  Contracts  for  the  survey  of  the  public  lands 
shall  not  become  binding  upon  the  United  States  until 
approved  by  the  Commissioner  of  the  General  Land 
Office,  except  in  such  cases  as  the  Commissioner  may 
otherwise  specially  order. 

12  Stat.  409 ;  R.  S.  2398.  Maguire  v.  Tyler,  1  Black,  201 ;  Parks  v.  Ross. 
11  How.  362-;  Spencer  r.  Lapsley,  20  id  264.  Reed  r.  Con  way,  26 
Mo.  13.  Decision  Sec.  lut.,  Feb.  27,  Io78. 

SEC.  112.  The  Commissioner  of  the  General  Land  Office 
has  power,  and  it  shall  be  his  duty,  to  fix  the  prices  per 
mile  for  public  surveys,  which  shall  in  no  case  exceed  the 
maximum  established  by  law;  and,  under  instructions  to 
be  prepared  by  the  Commissioner,  an  accurate  account 
shall  be  kept  by  each  surveyor-general  of  the  cost  of  sur- 


208  A    MANUAL    OF    LAND    SURVEYING. 

veying  and  platting  private  land  claims,  to  be  reported  to 
the  General  Land  Office,  with  the  map  of  such  claim; 
and  patents  shall  not  issue  for  any  such  private  claim, 
nor  shall  any  copy  of  such  survey  be  furnished,  until  the 
cost  of  survey  and  platting  has  been  paid  into  the  Treas- 
ury by  the  claimant  or  other  party;  and  before  any  land 
granted  to  any  railroad  company  by  the  United  States 
shall  be  conveyed  to  such  company  or  any  persons  entitled 
thereto,  under  any  of  the  acts  incorporating  or  relating  to 
said  company,  unless  such  company  is  exempted  by  law 
from  the  payment  of  such  cost,  there  shall  first  be  paid 
into  the  Treasury  of  the  United  States  the  cost  of  sur- 
veying, selecting,  and  conveying  the  same  by  the  said 
company  or  persons  in  interest. 

12  Stat.  4C9 ;  18  id.  384 ;  19  id.  Ill ;  E.  S.  2400     Railway  Co.  v.  Prescott, 

16  Wall.  6C3;  Railway  Co.  v.  McShane,  22  id.  444;  Hannewell  v. 

Cass  Co.,  22  id.  4G4;  Colorado  Co.  v.  Commissioners,  5  Otto,  259. 

Decisions  Sec.  Int.,  Dec.  17,  1874;  Feb.  27,  1873;  Feb.  20,  1879; 

March  5,  J879;  April  2,  1879.    Decisions  Com.  G.  L.  O.,  April  18, 

1867 ;  August  18,  1867;  Feb.  17, 1869 ;  March  26, 1870.    Cir.  G.  L.  O., 

June  2G,  1880. 

SEC.  113.  The  Commissioner  of  the  General  Land  Office 
may  authorize,  in  his  discretion,  public  lands  in  Oregon 
densely  covered  with  forests  or  thick  undergrowth,  to  be 
surveyed  at  augmented  rates,  not  exceeding  eighteen  dol- 
lars per  mile  for  standard  parallels,  fifteen  dollars  for 
townships,  and  twelve  dollars  for  section  lines;  and 
under  like  conditions  he  may  allow  augmented  rates  i;i 
California,  and  in  Washington  Territory,  not  exceeding 
eighteen  dollars  per  linear  mile  for  standard  parallels, 
sixteen  dollars  for  township,  and  fourteen  dollars  for 
section  lines. 

16  Stat.  304,  305 ;  17  id.  358 ;  R.  S.  2404,  2405.    Decision  Sec.  Int.,  June 
16,  1879.    Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  114.  Whenever  the  public  surveys,  or  any  portion 
of  them,  in  the  States  of  Oregon  and  California,  are  so 
required  to  be  made  as  to  render  it  expedient  to  make, 
compensation  for  the  surveying  thereof  by  the  clay  instead 


ORIGINAL    SURVEYS.  209 

of  by  the  mile,  it  shall  be  lawful  for  the  Commissioner  ot 
the  General  Land  Office,  under  the  direction  of  the  Secre- 
tary of  the  Interior,  to  make  such  fair  and  reasonable 
allowance,  as,  in  his  judgment,  may  be  necessary  to  insure 
the  accurate  and  faithful  execution  of  the  work. 

lOStat.  247;  R.  S.  2411.    Decision  Sec.  Int.,  June  16,  1879.    Cir.  G.  L. 
O.,  June  26, 1880. 

SEC.  118.  Each  surveyor-general,  when  thereunto  duly 
authorized  by  law,  shall  cause  all  confirmed  private  land 
claims  within  his  district  to  be  accurately  surveyed,  and 
shall  transmit  plats  ani  field-notes  thereof  to  the  Com- 
missioner of  the  General  Land  Office  for  his  approval. 
When  publication  of  such  surveys  is  authorizsd  by  law, 
the  proof  thereof,  together  with  any  objections  properly 
•  filed  and  all  evidence  submitted  either  in  support  of  or  in 
opposition  to  the  approval  of  any  such  survey,  shall  also 
be  transmitted  to  said  Commissioner. 

2  Stat.  326,  352;  3  id.  325 ;  5  id.  740 ;  9  id.  242,  633 ;  10  id.  244,  308,  599; 

11  id.  294;  12  id.  172,  209,  369,  409 ;  13  id.  332,  344;  14  id.  218;  16  id. 
64,  304 ;  18  id.  305;  19  id.  121,  202 :  R.  8.  2447.    Bissell  v.  Penrose,  8 
How.  317 ;  Villalobus  v.  TJ.  S.,  10  id.  541 ;  Ledoux  v.  Black,  18  id. 
473 ;  U.  S.  v.  Fossat,  20  id.  413;  Brown  «.  Huger,  21  id.  305 ;  U.  S.  r. 
Fossat,  21  id.  445 ,  Castro  r.  Hendricks,  23  id.  438;  Ballance  v.  For- 
syth,  24  id.  183;  tJ.  S.  v.  Sepulveda,  1  Wall.  104;  U.  S.  v.  Halleck, 
1  id,  439;  U.  S.  r.  Vallejo,  1  id.  658 ;  Sutter's  case  2  id.  562 ;  Fossat 
case,  2  id.  G49 ;  Higueras  v.  U.  £ ,  5  id.  827 ;  Alviso  v.  U.  S.,  8  id,  337. 

12  Op.  Att.  Gen.  116, 250;  14  id,  74,  601.    U.  S,  v.  Garcia,  1  Saw.  C.C. 
383;  Russell  v.  Henshaw,  1  id.  553;  Leroy  v.  Jamison,  3  id.  369; 
TJ.  S.  v.  Flint,  4  id.  42.    Dent  v.  Sergerson,  29  Mo.  480 ;  Fowler  v. 
Duvall,  11  La.  Ann.  5G1 ;  Waterman  v.  Smith,  13  Cal.  373;  Moore  v 
Wilkerson,  13  id.  478;  Men-it  v.  Judd,  14  id. CO;  Mott  v. Smith,  16id. 
534 ;  Johnson  v.  Van  Dyke,  20  id.  225 ;  McGarraghan  v.  Maxwell, 
27  id.  75;  Scale  v.  Ford,  29  id.  104.    Cir.  G.  L.  O.,  June  26, 1880. 

SEC.  120.  Every  person  who  in  any  manner,  by  threat 
or  force,  interrupts,  hinders,  or  prevents  the  surveying  of 
the  public  lands,  or  of  any  private  land  claim  which  has 
been  or  may  be  confirmed  by  the  United  States,  by  the 
persons  authorized  to  survey  the  same,  in  conformity 
with  the  instructions  of  the  Commissioner  of  the  General 

14 


210  A    MANUAL    OF    LAND    SURVEYING. 

Land  Office,  shall  be  fined  not  less  than  fifty  dollars  nor 
more  than  three  thousand  dollars,  and  be  imprisoned  not 
less  than  one  nor  more  than  three  years. 

4  Stat.  417 ;  E.  S.  2412. 

SEC.  121.  Whenever  the  President  is  satisfied  that  forci- 
ble opposition  has  been  offered,  or  is  likely  to  be  offered, 
to  any  surveyor  or  deputy  surveyor  in  the  discharge  of 
his  duties  in  surveying  the  public  lands,  it  may  be  lawful 
for  the  President  to  order  the  marshal  of  the  State  or 
district,  by  himself  or  deputy,  to  attend  such  surveyor 
or  deputy  surveyor  with  sufficient  force  to  protect  such 
officer  in  the  execution  of  his  duty,  and  to  remove  force 
should  any  be  offered. 

4  Stat.  417  ;  R.  S.  2413. 

SEC.  122.  The  President  is  authorized  to  appoint  sur- 
veyors of  public  lands,  who  shall  explore  such  vacant 
and  unappropriated  lands  of  the  United  States  as  produce 
the  live-oak  and  red- cedar  timbers,  and  shall  select  such 
tracts  or  portions  thereof,  where  the  principal  growth  is 
of  either  of  such  timbers,  as  in  the  judgment  of  the  Sec- 
retary of  the  Navy  may  be  necessary  to  furnish  for  the 
Navy  a  sufficient  supply  of  the  same.  Such  surveyors 
shall  report  to  the  President  the  tracts  by  them  selected, 
with  the  boundaries  ascertained  and  accurately  desig- 
nated by  actual  survey  or  water- courses. 

3  Stat.  347 ;  E.  S.  2459.  U.  S.  v.  Briggs,  9  How.  351. 

SEC.  123.  The  director  of  the  geological  survey  shall, 
under  the  Interior  Department,  have  the  direction  of  the 
geological  survey  and  the  classification  of  the  public 
lands  and  examination  of  the  geological  structure,  min- 
eral resources,  and  products  of  the  national  domain. 

20  Stat.  394. 

8.  Manner  of  Field  Work  and  Changes  that 
have  been  Made.— In  accordance  with  these  laws,  in- 
structions have  been  issued  from  time  to  time,  by  the 


ORIGINAL    SURVEYS.  211 

Commissioners  of  the  General  Land  Office,  directing  the 
manner  in  which  the  field  work  should  be  performed. 

In  the  earlier  surveys  under  the  act  of  1796  (Sec.  2395 
B.  S.  See  p.  199,  Sec.  99,  Third,)  the  township  was  sub- 
divided by  parallel  lines  two  miles  apart.  The  mile  posts 
were  planted  on  these  lines,  but  no  half  mile  (or  quarter- 
section)  corners  set. 

The  act  of  1800  provided  that  the  townships  west  of 
the  Muskingum  Kiver  should  be  subdivided  into  half 
sections  of  320  acres  each,  as  near  as  may  be,  by  parallel 
lines  run  through  them  from  east  to  west  and  from  north 
to  south  at  distances  of  a  mile  apart.  Half-mile  posts 
were  to  be  set  on  the  east  and  west  lines,  but  not  on  the 
lines  running  north  and  south. 

The  act  of  1805  (Sec.  2396  R.  S.  P.  200,  Sec.  100)  covers 
in  its  provisions  the  two  classes  of  surveys  above  noted, 
as  well  as  the  principles  governing  all  subsequent  surveys 
of  the  public  lands. 

Since  that  time,  few  changes  have  been  made  in  the 
manner  of  carrying  on  the  surveys. 

The  principal  changes  have  been  in  the  instruments 
used  and  in  the  manner  of  closing  the  eubdivisional 
lines  on  the  exterior  boundary  of  the  township. 

In  the  earlier  surveys,  the  lines  were  all  run  by  the 
magnetic  needle.  Now  the  direction  of  all  lines  must 
be  determined  independently  of  the  needle,  the  use 
of  which  for  running  lines  or  determining  courses  is 
prohibited. 

In  the  surveys  made  previous  to  1846,  the  deputy 
surveyors  were  required  to  close  the  subdivision  lines 
upon  the  corners  previously  set  on  the  east  line  of  the 
township,  but  not  on  those  set  on  the  north  and  west 
lines.  Double  corners  were  thus  produced  on  all  the 
exterior  lines  of  the  township.  The  same  system 


212  A    MANUAL    OF    LAND    SURVEYING. 

prevailed  in  some  of  th'e  surveying  districts  as  late  as 
1854,  and  perhaps  later.  It  is  thus  laid  down  in  the 
instructions  of  1815. 

"Each  side  of  a  section  must  be  made  one  mile  in 
measure  by  the  chain,  and  quarter-section  corners  are  to 
be  established  at  every  half  mile,  except  when  in  the 
closing  of  a  section  if  the  measure  of  the  closing  side 
should  vary  from  80  chains  or  one  mile,  you  are  in  that 
case  to  place  the  quarter- section  corners  equidistant,  or 
at  an  average  distance  from  the  corners  of  the  section; 
but  in  running  out  the  sectional  lines  on  the  west  or  north 
side  of  the  township,  you  will  establish  your  quarter- 
section  posts  or  corners  at  the  distance  of  half  a  mile 
from  the  last  corner,  and  leave  the  remaining  excess  or 
defect  on  the  west  or  north  tier  of  quarter-sections,  which 
balance  or  remainder  you  will  carefully  measure  and  put 
down  in  your  field-notes  in  order  to  calculate  the  remain- 
ing or  fractional  quarter-section  on  the  north  and  west 
side  of  the  township:  also  in  running  to  the  western  or 
northern  boundary,  unless  your  sectional  lines  fall  in  with 
the  posts  established  there  for  the  corners  of  sections  in 
the  adjacent  townships,  you  must  set  post  and  mark 
bearing  trees  at  the  points  of  intersection  of  your  lines 
with  the  town  boundaries,  and  take  the  distance  of  your 
corners  from  the  corners  of  the  sections  of  the  adjacent 
townships,  and  note  that  and  the  side  on  which  it  varies 
in  chains  or  links,  or  both. 

The  sections  must  be  made  to  close  by  running  a  ran- 
dom line  from  one  corner  to  another,  except  on  the  north 
and  west  ranges  of  sections,  and  the  true  line  between 
them  is  to  be  established  by  means  of  offsets." 

Under  the  present  system,  which  has  been  in  use  in 
some  parts  of  the  country  since  1846,  the  section  lines 
are  required  to  close  on  the  corners  previously  set  on  the 
north  and  west  boundaries,  the  same  as  on  the  east,  thus 
doing  away  with  the  system  of  double  section  corners. 


ORIGINAL    SURVEYS.  L'lo 

The  practice  in  the  several  surveying  districts  in  the 
United  States  does  not  seem  to  have  been  uniform  at  any 
time  previous  to  1860,  and  perhaps  not  always  since  that 
date.  For  instance,  in  the  Instructions  of  the  Commis- 
sioner of  the  General  Land  Office  to  surveyors-general, 
dated  Feb.  22,  1855,  which  is  stated  to  be  a  revision  of 
the  manual  of  surveying  instructions  prepared  for  Or- 
egon in  1851,  it  is  expressly  ordered  that  "double  corners 
are  to  be  nowhere  except  on  the  base  and  standard  lines;" 
while  in  the  instructions  to  deputy  surveyors  of  the 
United  States  for  the  district  of  Illinois  and  Missouri, 
published  in  1856,  P.  9,  the  deputy  surveyors  were 
directed  to  plant  their  closing  corners  at  the  intersec- 
tion of  their  lines  with  the  north  and  west  boundary  and 
return  their  direction  and  distance  from  the  corners  of 
the  corresponding  sections  on  the  north  and  west  of 
these  boundaries,"  the  surveyor-general  of  that  district 
thus  giving  different  instructions  from  those  of  the 
Commissioner  of  the  General  Land  Office. 

9.  Fractional  Areas..— It  has  been  a  puzzle  to 
many  surveyors  to  know  how  the  area  of  the  fractional 
quarter-sections  adjoining  the  north  and  west  boundaries 
of  the  township  were  calculated.  It  has  been  just  as 
much  of  a  puzzle  to  the  surveyors-general  and  Commis- 
sioners of  the  General  Land  Office. 

Edward  Tiffin,  surveyor -general  of  the  Northwest 
Territory,  in  1815  issued  instructions  how  to  do  it,  which 
instructions  were  made  applicable  to  the  surveys  in  Ohio, 
Michigan,  Arkansas  and  Missouri.  Under  these  instruc- 
tions, the  calculations  of  the  areas  of  these  fractions 
were  to  be  made  on  the  assumption  that  the  quarter- 
posts  on  the  township  and  range  lines  were  common  to 
the  sections  on  both  sides  of  these  lines,  thus  making 
the  lengths  of  the  fractions  more  or  less  unequal  where 
there  wrere  double  section  corners.  This  plan  does  not 
seem  to  have  been  in  force  long,  or  to  have  been  very 
generally  followed.  Another  plan  quite  extensively 
adopted  was  to  make  the  calculations  on  the  theory  that 
all  the  north  and  south  quarter-lines  of  these  fractional 
sections  were  to  be  parallel  with  the  east  line  of  the  sec- 


214  A    MANUAL    OF    LAND    SURVEYING. 

tions,  and  all  east  and  west  quarter-lines  parallel  with  the 
south  line  of  the  sections.  Neither  plan  was  in  harmony 
with  the  law  of  1805,  which  required  "the  corners  of  half 
and  quarter  sections  not  marked  on  the  surveys  to  be 
placed  as  nearly  as  possible  equidistant  from  those  two 
corners  which  stand  on  the  same  line." 

The  plan  under  which  most  if  not  all  the  fractional 
areas  of  Michigan  were  calculated  was  on  the  theory  that 
the  quarter-posts  on  the  township  and  range  lines  were  to 
be  placed  midway  between  their  respective  section  corners. 

Previous  to  1828,  the  deputy  surveyors  were  required  to 
return  with  their  field  notes  plats  of  all  the  townships 
which  they  surveyed,  and  to  calculate  the  area  of  the 
fractions.  These  plats  were  rudely  constructed,  and  in 
many  cases  the  areas  put  down  on  them  were  erroneous. 
If  this  was  found  out  before  the  land  was  sold,  the  areas 
were  re  calculated  in  the  surveyor-general's  office.  In 
making  the  calculations  of  the  areas  of  the  fractions 
along  the  township  and  range  lines,  some  of  the  deputies 
considered  the  quarter-section  corners  along  those  lines 
as  common  to  the  sections  on  both  sides,  some  adopted 
the  second  method  described  above,  while  the  areas  of 
many  of  the  fractions  appear  to  have  been  put  down 
without  any  calculation  whatever. 

In  the  U.  S.  Surveying  Instructions  of  Jan.  1,  1902, 
the  following  rules  are  given  :  — 

In  the  north  tier  of  Sections  the  fractional  lots  along 
the  boundary  are  numbered  1  to  4  from  east  to  west. 
In  the  west  tier  they  are  numbered  from  north  to 
south.  In  Section  6  they  are  numbered  from  1  to  7 
from  the  N.  E.  corner  of  the  Section  along  the  boundary 
to  the  S.  W.  corner. 

1.  In  regular  townships,  the  tracts  of  land  in  each  sec- 
tion adjoining  the  north  and  west  boundaries  of  such 
townships,  in  excess  of  the  regularly  subdivided  480 
acres  (except  in  section  6),  will,  in  general,  be  in  the 
form  of  trapezoids,  80.00  chains  in  length  by  about  20 
chains  in  width. 

On  the  plats  of  such  townships,  each  of  said  tracts 
will  be  divided  into  four  lots,  by  drawing  broken  lines 


ORIGINAL    SURVEYS.  215 

at  intervals  of  20.00  chains,  parallel  to  the  ends  of  the 
tracts,  which  will  be  regarded  as  parallel  to  each  other. 

With  the  exception  of  section  6.  the  south  boundaries 
of  sections  of  the  north  tier,  when  within  prescribed 
limits,  will  be  called  80.00  chains. 

When  the  above-named  conditions  obtain,  the  areas 
of  the  lots  in  any  one  tract  (except  in  section  6)  may  be 
determined,  as  follows:  — 

Divide  the  difference  between  the  widths  of  the  ends 
of  the  tract  by  4:  if  3  remains,  increase  the  hundredth 
figure  of  the  quotient  by  a  unit;  in  all  other  cases  disre- 
gard the  fraction:  call  the  quotient  thus  obtained,  "d  :*' 
then,  taking  the  end  widths  of  the  tract  in  chains  and 
decimals  of  a  chain,  the  areas  of  the  lots,  in  acres,  will  be: — 

Of  the  smallest  lot:  twice  the  width  of  the  lesser  end, 
pfar**d;M 

Of  the  largest  lot:  twice  the  width  of  the  greater  end, 
minus  "d;  " 

Of  the  smaller  middle  lot:  sum  of  the  widths  of  the 
ends,  minus  "d: " 

Of  the  larger  middle  lot:  sum  of  the  widths  of  the  ends, 
pfc^"d." 

A  check  on  the  computation  may  be  had  by  multiply- 
ing the  sum  of  the  widths  of  the  ends  of  the  tract  by  4; 
the  product  should  agree  exactly  with  the  total  area  of 
the  four  lots. 

The  proper  application  of  the  above  rules  will  always 
give  areas  correct  to  the  nearest  hundredth  of  an  acre;  and, 
as  the  use  of  fractions  is  entirely  avoided,  the  method 
is  recommended  for  its  simplicity  and  accuracy. 

Example  1. 

The  i  difference  of  latitudinal  boundaries  is  0.031 
chains;  consequently,  "d  "  is  .04  chains;  then, 

18.35X  2  +.04=  36.74  acres,  the  area  of  lot  1 ; 
18.50X  2  —.04=  36.96  acres,  the  area  of  lot  4; 
18.50+18.35  —.04=  36.81  acres,  the  area  of  lot  2: 
18.50+18.35  +.04=  36.89  acres,  the  area  of  lot  3 : 
Check:  [18.35+18 .50]  x  4=147.40  acres,  tlie  area  of  the  four  lots. 

The  arithmetical  operations  are  here  written  in  de- 
tail, for  the  purpose  of  illustration;  but  the  practical 
computer  will  perform  all  the  work  mentally. 


216  A    MANUAL    OF    LAND    SURVEYING. 

2.  Section  6.  The  areas  of  lots  5,  6,  and  7  may  be  ob- 
tained by  the  foregoing  rules  in  all  cases,  except  when 
the  township  closes  on  a  base  line  or  standard  par- 
allel; also,  the  area  of  lot  4,  provided  both  meridional 
boundaries  are  80.00  chains  in  length;  when  the  last 
condition  obtains,  the  areas  of  lots  1,  2,  and  3  will  be 
equal,  and  each  will  contain  40.00  acres. 

In  any  case  where  the  west  boundary  of  sec.  6,  is  80.00 
chains,  and  the  east  boundary  either  greater  or  less  than 
80.00  chains,  the  areas  of  lots  1,  2,  3,  and  4  will  be  com- 
puted as  follows:  — 

Determine  the  difference,  "q,"  between  the  east 
boundaries  of  lots  1  and  4  by  the  following  propor- 
tion:— 

N.  bdy.  sec.  6.:  diff.  of  meridional  bdrs.  sec.  6.  ::60chs. : 
q;  then  will  E.  bdy.  lot  4— E.  bdy.  lot  l=bq;  in  which, 
"q  "  will  be  added  when  the  east  boundary  of  sec.  6  is 
less  than  80.00  chains;  but  subtracted  when  said  east  bound- 
ary is  greater  than  80.00  chains. 

Now  take  one  third  of  "q,"  and  add  it  to  the  shorter 
east  boundary  of  lots  1  or  4,  as  conditions  may  require, 
and  thereby  determine  the  length  of  one  of  the  meridi- 
onal boundaries  of  lot  2;  to  which  again  add  "one 
third  of  q,"  and  thus  obtain  the  length  of  the  opposite 
side  of  lot  2.  The  areas  of  lots  1,  2,  and  3,  in  acres,  will 
be  found  by  taking  the  sum  of  their  respective  meridi- 
onal boundaries,  expressed  in  chains  and  decimals  of  a  chain. 

The  area  of  lot  4  may  be  had  by  multiplying  its  mean 
width  by  its  mean  length. 

Finally,  to  test  the  entire  work,  multiply  the  sum  Of 
the  latitudinal  boundaries  by  4,  and  to  the  product  add 
the  area  of  the  small  triangle  C  A  B,  if  the  east  boun- 
dary is  greater  than  80.00  chains;  but  subtract  the  area  of 
said  small  triangle  if  the  east  boundary  is  less  than  80.00 
chains.  These  operations,  correctly  performed,  will 
give  the  true  area  of  the  section,  which  should  agree  exactly 
with  the  total  area  of  its  legal  subdivisions,  obtained  as 
directed  in  the  preceding  paragraphs. 

Example  2. 

Compute  areas  of  lots  5,  6,  and  7  of  sec.  6,  as  directed 


ORIGINAL    PURVEYS.  217 

in  paragraph  1,  and  illustrated  by  the  example:  then 

write:  — 

chs.     chs.       chs.        chs.  chs. 

77.75  :  0.05  ::  60.00  :  0.0386=q;  ^  q=0.0129 

chs.       chs.      chs. 

20.0500— 0.0386=20.01.  the  E.  bdy.  of  lot  4; 
20.01 14-f-O.Ol 29 =20.02,  the  E.  bdy.  of  lot  3: 
20.0243+0.0129=20.04,  the  E.  bdy.  of  lot  2'. 

Then,  for  the  areas  of  lots  1,  2,  3,  and  4,  we  have:— 

chs.      chs.  acres. 

20.05-1-20.04 =  40.09,  the  area  of  lot  1 ; 

20.U4-j-20.02 =  40.06,  the  area  of  lot  2 : 

20.02-j-20.01   =  40.03,  the  area  of  lot  3 ; 

=  35.54,  the  area  of  lot  4. 


2  2 

Also  [17.78+17.S7]  X3  =  106.95.'the  area  of  lots  5,  6,  and  7. 

Area  of  regular  subdivisions =360. 00 

Total =622.67.  the  area  of  Sec.  6. 


chs.      chs. 
Check:  [77.87+77.75] X4=622.48 

77.75X  0.025      =    0.19,  the  area  of  triangle  CAB.    ' 

Total  =622.67,  which  agrees  with  the  area  of  section  6, 

before  determined. 

3.  The  area  in  acres  of  a  tract  40.00  chains  long,  ad- 
joining north  or  west  township  boundaries  (except  in 
"N".  W.  i  sec.  6),  is  equal  to  the  sum  of  its  paralkl  bound~ 
aries  (expressed  in  chains  and  decimals  thereof)  multi- 
plied by  2 ;  (e.  g.}  the  area  of  lots  6  and  7,  is  [17.87-j-17.81] 
X2=71.36  acres. 

The  area  in  acres  of  a  tract  60.00  chains  long,  situated 
as  above  described  (excluding  lot  4,  of  sec.  6),  may  be 
found  by  multiplying  the  sum  of  its  parallel  "boundaries 
(expressed  in  chains  and  decimals  of  a  chain)  by  3;  (e.  g.) 
Fig.  6 ;  south  boundary  lot  4=17.78  chs.;  area  of  lots  5,  6, 
and  7  is  [1 7. 78+17. 87] X 3=106. 95  acres.  (See  example  2.) 

The  area  in  acres  of  quarter  sections  adjoining  north 
and  west  township  boundaries  (excluding  N.  W.  i  sec. 
6),  may  be  obtained  by  multiplying  the  sum  of  their 
parallel  boundaries  (taken  in  chains  and  decimals  of  a 
chain),  by  2;  (e.  g.}  the  area  of  S.  W.  i  sec.  6  (Fig.  6),  is 
[37.87+37.81] x 2=151.36  acres. 

The  area  in  acres  of  any  section  along  the  north  and 
west  boundaries  of  regular  townships  (except  sec.  6)  may 


218  A    MANUAL    OF    LAND    SURVEYING. 

be  had  by  multiplying  the  sum  of  its  parallel  boundaries 
(expressed  in  chains  and  decimals  of  a  chain)  by  4;  (e.  g.) 
the  area  of  sec.  1  (Plate  IV)  is  [80.00-|-79.77]x4=:639.08 
acres. 

The  area  in  acres  of  a  theoretical  township  may  be  ob- 
tained by  multiplying  the  sum  of  its  latitudinal  bound- 
aries (expressed  in  chains  and  decimals  of  a  chain)  by  24 
(e.  g.)  the  area  of  a  township  is  [480. 00-f  479. 34} x 24=23, 
024.16  acres. 

10.  Instructions  of  i9o2.  —  The  U.  S.  Manual  of 
Surveying  Instructions  for  1902,  is  a  large  volume  of 
203  pages,  and  contains  minute  instructions  in  regard 
to  all  the  operations  of  the  survey  of  the  public  lands 
and  private  land  claims.  It  is  furnished  to  Deputy 
U.  S.  Surveyors  and  may  be  had  by  others  who  apply 
for  it  to  the  Commissioner  of  the  General  Land  Of- 
fice at  Washington.  The  following  extracts  are  made 
from  it :  — 

SYSTEM  OF  RECTANGULAR  SURVEYING. 

1.  Existing  law  requires  that  in  general  the  public 
lands  of  the  United  States  "shall  be  divided  by  north 
and  south  lines  run  according  to  the  true  meridian,  and 
by  others  crossing  them  at  right  angles  so  as  to  form 
townships  six  miles  square,"  and  that  the  corners  of  the 
townships  thus  surveyed  "must  be  marked  with  pro- 
gressive numbers  from  the  beginning." 

Also,  that  the  townships  shall  be  subdivided  into 
thirty-six  sections,  each  of  which  shall  contain  six  hun- 
dred and  forty  acres,  as  nearly  as  may  be,  by  a  system  of 
two  sets  of  parallel  lines,  one  governed  by  true  meridi- 
ans and  the  other  by  parallels  of  latitude,  the  latter  in- 
tersecting the  former  at  right  angles,  at  intervals  of  a 
mile. 

2.  In  the  execution  of  the  public  surveys  under  exist- 
ing law,  it  is  apparent  that  the  requirements  that  the 
lines  of  survey  shall  conform  to  true  meridians,  and  that 
the  townships  shall  be  6  miles  square,  taken  together, 
involve  a  mathematical  impossibility  due  to  the  con- 
vergency  of  the  meridians. 


ORIGINAL    SURVEYS.  219 

Therefore,  to  conform  the  meridianal  township  lines 
to  the  true  meridians  produces  townships  of  a  trape- 
zoidal form  which  do  not  contain  the  precise  area  of 
23.040  acres  required  by  law,  and  which  discrepancy  in- 
creases with  the  increase  in  the  convergency  of  the 
meridians,  as  the  surveys  attain  the  higher  latitudes. 

In  view  of  these  facts,  and  under  the  provisions  of 
section  2  of  the  act  of  May  18,  1796,  that  sections  of  a 
mile  square  shall  contain  640  acres,  as  nearly  as  may  6e, 
and  also  under  those  of  section  3  of  the  act  of  May  10, 
1800,  that  "in  all  cases  where  the  exterior  lines  of  the 
townships,  thus  to  be  subdivided  into  sections  and  half 
sections,  shall  exceed,  or  shall  not  extend  6  miles,  the 
excess  or  deficiency  shall  be  specially  noted,  and  added 
to  or  deducted  from  the  western  or  northern  ranges  of 
sections  or  half  sections  in  such  township,  according  as 
the  error  may  be  in  running  lines  from  east  to  west,  or 
from  south  to  north ;  the  sections  and  half  sections 
bounded  on  the  northern  and  western  lines  of  such 
townships  shall  be  sold  as  containing  only  the  quantity 
expressed  in  the  returns  and  plats,  respectively,  and  all 
others  as  containing  the  complete  legal  quantity."  the 
public  lands  of  the  United  States  shall  be  surveyed  un- 
der the  methods  of  the  system  of  rectangular  surveying, 
which  harmonizes  the  incompatibilities  of  the  require- 
ments of  law  and  practice,  as  follows:  — 

First.  The  establishment  of  a  principal  meridian  con- 
forming to  the  true  meridian,  and,  at  right  angles  to  it, 
a  base  line  conforming  to  a  parallel  of  latitude. 

Second.  The  establishment  of  standard  parallels  con- 
forming to  parallels  of  latitude,  initiated  from  the 
principal  meridian  at  intervals  of  24  miles  and  extended 
east  and  west  of  the  same. 

Third.  The  establishment  of  guide  meridians  con- 
forming to  true  meridians,  initiated  upon  the  base  line 
and  successive  standard  parallels  at  intervals  of  24  miles, 
resulting  in  tracts  of  land  24  miles  square,  as  nearly  as 
may  he,  which  shall  be  subsequently  divided  into  tracts 
of  land  6  miles  square  by  two  sets  of  lines,  one  conform- 
ing to  true  meridians,  crossed  by  others  conforming  to 


220  A    MANUAL    OF    LAND    SURVEYING. 

parallels  of  latitude  at  intervals  of  6  miles,  containing 
23,040  acres,  as  nearly  as  may  be,  and  designated  townships. 
Such  townships  shall  be  subdivided  into  thirty-six 
tracts,  called  sections,  each  of  which  shall  contain  640 
acres,  as  nearly  as  may  be,  by  two  sets  of  parallel  lines, 
one  set  parallel  to  a  true  meridian  and  the  other  conforming 
to  parallels  of  latitude,  mutually  intersecting  at  intervals 
of  1  mile  and  at  right  angles,  as  nearly  as  may  be. 

Any  series  of  contiguous  townships  situated  north 
and  south  of  each  other  constitutes  a  range,  while  such 
a  series  situated  in  an  east  and  west  direction  consti- 
tutes a  tier. 

By  the  terms  of  the  original  law,  and  by  general 
practice,  section  lines  were  surveyed  from  south  to 
north  and  from  east  to  west,  in  order  to  uniformly 
place  excess  or  deficiency  of  measurement  on  the  north 
and  west  sides  of  the  townships.  But  under  modern 
conditions  many  cases  arise  in  which  a  departure 
from  this  method  is  necessary.  Where  the  west  or 
the  north  boundary  is  sufficiently  correct  as  to  course, 
to  serve  as  a  basis  for  rectangular  subdivision,  and 
the  opposite  line  is  defective,  the  section  lines  should 
be  run  by  a  reversed  method. 

For  convenience  the  well-surveyed  lines  on  which 
subdivisions  are  to  be  based,  will  be  called  govern- 
ing boundaries  of  the  township. 

3.  The  tiers  of  townships  will  be  numbered,  to  the 
north  or  south,  commencing  with  No.  1,  at  the  base 
line;  and  the  ranges  of  the  townships,  to  the  east  or 
west,  beginning  with  No.  1,  at  the  principal  meridian 
of  the  system. 

4.  The  thirty-six  sections  into  which  a  township  is 
subdivided  are  numbered,  commencing  with  number 
one  at  the  northeast  angle  of  the  township,  and  proceed- 
ing west  to  number  six,  and  thence  proceeding  east  to 
number  twelve,  and  so  on,  alternately,  to  number  thir- 
ty-six in  the  southeast  angle.    In  all  cases  of  surveys  of 
fractional  townships,  the  sections  will  bear  the  same 
numbers  they  would  have  if  the  township  was  full, and 
where  doubt  arises  as  to  which  section  numbers  should 


ORIGINAL    SURVEYS.  221 

be  omitted,  the  proper  section  numbers  will  be  used 
on  the  side  or  sides  which  are  governing  boundaries, 
leaving  any  deficiency  to  fall  on  the  opposite  sides. 

5.  Standard  parallels,  formerly  called  correction 
lines,  shall  be  established  at  intervals  of  every  24 
miles,  north  and  south  of  the  base  line,  and  guide 
meridians  at  intervals  of  every  24  miles,  east  and 
west  of  the  principal  meridian;  thus  confining  the  er- 
rors resulting  from  convergence  of  meridians  and  inac- 
curacies in  measurement  within  comparatively  small 
areas. 

Instruments. —  6.  The  surveys  of  the  public  lands 
of  the  United  States,  embracing  the  establishment  of 
base  lines,  principal  meridians,  standard  parallels,  rnean- 
iler  lines,  and  the  subdivisions  of  townships,  will  be 
made  with  instruments  provided  with  the  accessories 
necessary  to  determine  a  direction  with  reference  to  the 
true  meridian,  independently  of  the  magnetic  needle. 

Burt's  improved  solar  compass,  or  a  transit  of  ap- 
proved construction,  with  or  without  solar  attachment, 
will  be  used  in  all  cases.  When  a  transit  without  solar 
attachment  is  employed,  Polaris  observations  and  the  re- 
tracements  necessary  to  execute  the  work  in  accordance 
with  existing  law  and  the  requirements  of  these  instruc- 
tions will  be  insisted  upon.  Observations  every  clear 
night  will  be  necessary  to  secure  accuracy  in  the  di- 
rection of  transit  reference  lines,  when  solar  appa- 
ratus is  not  used.  The  method  of  connecting  surveys 
with  the  stellar  meridian  should  distinctly  appear  in 
the  field  notes,  as  evidence  that  the  courses  were  not 
derived  from  the  magnetic  needle. 

7.  Deputies  using  instruments  with  solar  apparatus 
will  be  required  to  make  observations  on  the  star  Polaris 
at  the  beginning  of  every  survey,  and,  whenever  necessary, 
to  test  the  accuracy  of  the  solar  apparatus. 

The  observations  required  to  test  the  adjustments  of 
the  solar  apparatus  will  be  made  at  the  corner  where 
the  survey  begins,  or  at  the  camp  of  the  deputy  surveyor 
nearest  said  corner ;  and  in  all  cases  the  deputy  wiU 


222  A    MANUAL    OF    LAND    SUBVEYING. 

fully  state  in  the  field  notes  the  exact  locatforr  of  the 
observing  station. 

Deputy  surveyors  will  examine  the  adjustments  of 
their  instruments,  and  take  the  latitude  daily,  weather  per- 
mitting, while  running  all  lines  of  the  public  surveys.  They 
will  make  complete  records  in  their  field  notes,  under 
proper  dates,  of  the  making  of  all  observations  in  com- 
pliance with  these  instructions,  showing  the  character 
and  condition  of  the  instrument  in  use,  and  the  precis- 
ion attained  in  the  survey,  by  comparing  the  direction 
of  the  line  run  with  the  meridian  determined  by  obser- 
vation. 

On  every  survey  executed  with  solar  instruments,  the 
deputy  will,  at  least  once  on  each  working  day,  record  in  his 
field  notes  the  proper  reading  of  the  latitude  arc;  the 
declination  of  the  sun,  corrected  for  refraction,  set  off 
on  the  declination  arc;  and  note  the  correct  local  mean 
time  of  his  observation,  which,  for  the  record,  will  be 
taken  at  least  two  hours  from  apparent  noon. 

In  field  inspection  of  contract  surveys,  the  exam- 
iners are  required  to  obtain  the  meridian,  both  by 
solar  and  stellar  observations,  testing  their  instru- 
ments fully  before  reporting  on  the  courses  of  the 
deputy's  lines.  Hence  no  deputy  should  incur  risk 
by  omitting  any  of  the  safeguards  here  required  as 
essential  to  accurate  work. 

8.  The  construction  and  adjustments  of  all  surveying 
instruments  used  in  surveying  the  public  lands  of  the 
United  States  will  be  tested  at  least  once  a  year,  and 
oftener,  if  necessary,  on  the  true  meridian,  established 
under  the  direction  of  the  surveyor  general  of  the  dis- 
trict; and  if  found  defective,  the  instruments  shall  un- 
dergo such  repairs  or  modifications  as  may  be  found 
necessary  to  secure  the  closest  possible  approximation 
to  accuracy  and  uniformity  in  all  field  work  controlled 
by  such  instruments. 


ORIGINAL    SURVEYS.  223 

9.  Chaining. — The  instruments  for  measuring 
lines  are  the  chain  and  pins.  Each  deputy  will  be  pro- 
vided with  a  standard  steel  chain  or  steel  tape  of  ap- 
proved style,  precisely  ad  justed  to  the  standard  meas- 
ures kept  by  the  surveyor  general.  The  deputy's 
standard  measure  will  not  be  used  on  the  field  work, 
but  be  carefully  preserved  in  camp  and  used  for  pur- 
poses of  frequent  comparison  with  his  field  chains  or 
steel  tapes,  in  order  that  changes  due  to  constant  use 
may  be  discovered  at  the  beginning  of  each  day's 
work.  All  his  returns  of  distance  will  be  made  in 
miles,  chains,  and  links,  a  chain  of  100  links  being 
equal  to  66  feet.  Engineers'  chains  reading  by  feet 
only  are  not  to  be  used  in  public  land  surveys.  Dis- 
tances of  height  or  depth  may  be  given  in  feet  or 
inches.  In  these  details  the  specimen  field  notes  are 
to  be  observed. 

The  simple  conditions  imperatively  demanded  for 
all  accurate  measurements  are  specified  in  the  chain- 
man's  oath,  promising  that  he  will  level  the  chain 
upon  even  and  uneven  ground,  will  plumb  the  pins, 
either  by  sticking  or.  dropping  them,  and  will  re- 
port the  true  distances.  These  brief  rules,  faith- 
fully observed,  will  render  chaining  sufficiently  exact 
to  stand  the  test  of  inspection  by  strict  examiners. 

Before  chainmen  are  entrusted  with  their  actual 
duties,  they  should  be  exercised  for  practice  and 
thoroughly  instructed,  under  the  eye  of  their  em- 
ployer, by  chaining  two  or  three  times  over  one  or 
more  trial  lines  of  hilly  or  mountainous  surface,  to 
ascertain  the  accuracy  and  uniformity  of  the  results. 
The  methods  used  by  competent  surveyors  to  obtain 
true  horizontal  distance  over  steep  slopes,  are  too 
important  to  be  disregarded,  yet  too  elementary  to  be 
given  here.  When  using  only  a  portion  of  the  chain, 
on  steep  hill-sides,  especially  in  a  strong  wind,  ac- 
curacy requires  a  plumb-line  or  some  equivalent 
means,  to  mark  the  vertical.  The  dropping  of  flagged 
pins  not  loaded,  too  often  in  such  cases  leads  to  re- 
peated and  serious  error,  which  may  be  avoided  by 


224  A    MANUAL    OF    LAND    SURVEYING. 

dropping  a  more  suitable  object,  such  as  a  piece  of 
metal  carried  in  the  pocket. 

If  any  other  methods  of  obtaining  measurements 
up  or  down  hills  or  across  ravines  be  resorted  to,  ex- 
cept that  here  authorized,  the  facts  will  be  stated  in 
the  returns,  and  the  distances  must  well  sustain  the 
tests  of  the  field  examiner. 

10.  Marking  Line.— The  marking  of  trees  and 
brush  along  lines  was  required  by  law  as  positively  as 
the  erection  of  monuments,  by  the  act  of  1796,  which 
is  still  in  force.  The  old  rules  therefor  are  unchanged. 

All  lines  on  which  are  to  be  established  the  legal 
corner  boundaries  will  be  marked  after  this  method, 
viz  :  Those  trees  which  may  be  intersected  by  the  line 
will  have  two  chops  or  notches  cut  011  the  sides  facing 
the  line,  without  any  other  marks  whatever.  These 
are  called  sight  trees,  or  line  trees.  A  sufficient  num- 
ber of  other  trees  standing  within-50  links  of  the  line, 
on  either  side  of  it,  will  be  blazed  on  two  sides  di- 
agonally or  quartering  toward  the  line,  in  order  to 
render  the  line  conspicuous,  and  readily  to  be  traced 
in  either  direction,  the  blazes  to  be  opposite  each 
other,  coinciding  in  direction  with  the  line  where  the 
trees  stand  very  near  it,  and  to  approach  nearer  each 
other  toward  the  line,  the  farther  the  line  passes  from 
the  blazed  trees.  In  early  surveys,  an  opposite 
practice  prevailed 

Due  care  will  ever  be  taken  to  have  the  lines  so 
well  marked  as  to  be  readily  followed,  and  to  cut  the 
blazes  deep  enough  to  leave  recognizable  scars  as  long 
as  the  trees  stand.  This  can  be  attained  only  by 
blazing  through  the  bark  to  the  wood.  Trees  marked 
less  thoroughly  will  not  be  considered  sufficiently 
blazed.  Where  trees  two  inches  or  more  in  diame- 
ter occur  along  a  line,  the  required  blazes  will  not 
be  omitted. 

Lines  are  also  to  be  marked  by  cutting  away 
enough  of  the  undergrowth  of  bushes  or  other  vege- 
tation to  facilitate  correct  sighting  of  instruments. 
Where  lines  cross  deep  wooded  valleys,  by  sighting 


ORIGINAL    SURVEYS.  225 

over  the  tops,  the  usual  blazing  of  trees  in  the  low 
ground  when  accessible  will  be  performed,  that  set- 
tlers may  find  their  proper  limits  of  land  and  timber 
without  resurvey. 

The  practice  of  blazing  a  random  line  to  a  point 
some  distance  away  from  an  objective  corner,  and 
leaving  through  timber  a  marked  line  which  is  not 
the  true  boundary,  is  unlawful,  and  no  such  surveys 
are  acceptable.  The  decisions  of  some  State  courts 
make  the  marked  trees  valid  evidence  of  the  place  of 
the  legal  boundary,  even  if  such  line  is  crooked,  and 
has  the  quarter-section  corner  far  off  the  blazed  line. 

On  trial  or  random  lines,  therefore,  the  trees  will 
not  be  blazed,  unless  occasionally,  from  indispen- 
sable necessity,  and  then  it  will  be  done  so  guardedly 
as  to  prevent  the  possibility  of  confounding  the 
marks  of  the  trial  line  with  the  true.  But  bushes 
and  limbs  of  trees  may  bo  lopped,  and  stakes  set  on 
the  trial  or  random  line,  at  every  ten  chain*,  to 
enable  the  surveyor  on  his  return  to  follow  and  cor- 
ro(  t  the  trial  line  and  establish  therefrom  the  true 
line.  To  prevent  confusion,  the  temporary  stakes  set 
on  the  trial  or  random  line  will  be  removed  when  the 
surveyor  returns  to  establish  the  true  line. 

The  terms  of  each  act  making  appropriation  for 
compensation  of  surveys,  allow  increased  pay  for 
lines  passing  through  lands  "  covered  with  dense  un- 
dergrowth." The  evident  purpose  of  the  increase  is 
to  compensate  the  surveyor  for  the  additional  labor 
and  delay  of  cutting  away  brush  and  trees  which  ob- 
struct the  proper  survey  of  the  line,  and  also  of 
blazing  the  line  as  required  by  law. 

By  dense  undergrowth  is  meant  thick  bushes, 
boughs,  or  other  vegetable  growth  of  such  height  as 
to  obstruct  the  use  of  the  transit  and  require  cutting 
away  to  obtain  sights  along  line  ;  also  bushes,  brush, 
or  vines,  that  are  of  such  character  as  to  seriously 
impede  the  work  of  traversing  and  chaining  the  line. 

Increased  rates  for  heavy  timber  or  dense  under- 
growth will  not  be  allowed  for  lines  on  which  no 


220  A    MANUAL    OF    LAND    SURVEYING. 

cutting  away  of  brush  is  done  or  is  necessary,  or 
where  blazing  of  timber  is  generally  neglected,  if 
these  conditions  shall  be  shown  by  field  inspection. 

Insuperable  Objects  on  Line  — Witness 
Points.— 1.  Under  circumstances  where  the  survey  of 
a  line  is  obstructed  by  an  impassable  obstacle,  such  as  a 
pond,  swamp,  or  marsh  (not  meanderable),  the  line  will 
be  prolonged  across  such  obstruction  by  making  the 
necessary  right-angle  offsets;  or,,  if  such  proceeding  is 
impracticable,. a  traverse  line  will  be  run,  or  some  proper 
trigonometical  operation  will  be  employed  to  locate  the 
line  on  the  opposite  side  of  the  obstruction;  and  in  case 
the  line,  either  meridional  or  latitudinal,  thus  regained, 
is  recovered  beyond  the  intervening  obstacle,  said  line 
will  be  surveyed  back  to  the  margin  of  the  obstruction 
and  all  the  particulars,  in  relation  to  the  field  operations,  will 
be  fully  stated  in  the  field  notes. 

2.  As  a  guide  in  alinement  and  measurement,  at  each 
point  where  the  line  intersects  the  margin  of  an  obstacle, 
a  witness  point  will  be  established,  except  when  such  point 
is  less  than  20  chains  distant  from  the  true  point  for  a 
legal  corner  which  falls  in  the  obstruction,  in  which 
case  a  witness  corner  will  be  established  at  the  intersec- 
tion. 

3.  In  a  case  where  all  the  points  of  intersection  with 
the  obstacle  to  measurement  fall  more  than  20  chains  from 
the  proper  place  for  a  legal  corner  in  the  obstruction, 
and  a  witness  corner  can  be  placed  on  the  offset  line 
within  20  chains  of  the  inaccessible  corner  point,  such 
"  witness  corner  "  will  be  established. 

Establishing  Corners.— 1.  To  procure  the  faith- 
ful execution  of  this  part  of  a  surveyor's  duty,  is  a  mat- 
ter of  the  utmost  importance.  After  true  coursing  and 
most  exact  measurements,  the  establishment  of  corners 
is  the  consummation  of  the  field  work.  Therefore,  if 
the  corners  be  not  perpetuated  in  a  permanent  and 
workmanlike  manner,  the  principal  object  of  surveying 
operations  will  not  have  been  attained. 

2.  The  points  at  which  corners  will  be  established  are 


ORIGINAL    SURVEYS.  227 

fully  stated  in  the  several  articles:  "Base  Lines," 
"Principal  Meridians."  "Standard  Parallels,"  etc., 
following  the  title  "  Initial  Points." 

All  marking  of  letters  and  figures  should  be  done 
neatly,  distinctly,  and  durably,  using  the  tools  best 
adapted  to  the  purpose,  and  keeping  them  in  good 
order.  These  tools  are  the  chisel  and  hammer  for 
marking  stones,  and  the  scribing-tool  or  gouge  for 
surfaces  of  wood.  Since  the  greatest  permanency 
requires,  stone  corner  monuments,  and  the  perishable 
nature  of  wood  prohibits  its  use  where  stones  can  be 
found  or  brought,  the  deputy  should  be  provided 
with  good  chisels,  to  enable  him  to  mark  neatly  and 
expeditiously,  using  arabic  figures  for  all  numbers. 

Surveying  Monuments  —  1.  These  consist  of 
what  is  called  the  corner,  and  its  accessories.  The 
corner  itself  should  be  durable  and  firmly  imbedded. 
It  may  consist  of  an  iron  monument,  rod,  or  pipe,  a 
cross  cut  on  a  ledge,  or  a  marked  stone ;  or  in  case 
these  can  not  be  obtained,  then  a  post  of  durable 
timber.  Where  a  stone  corner  has  to  be  set  upon  a 
ledge  of  surface  rock,  it  should  be  of  large  size  and 
supported  in  a  well-built  stone  mound,  with  its  marks 
well  shown  ;  in  addition  to  which,  the  usual  witness 
mound  will  be  separately  built. 

Descriptions  of  Corners.— 1.  The  form  and  lan- 
guage used  in  the  following  articles,  in  describing,  for 
each  one  of  the  thirteen  classes  of  corners,  eight  specific 
constructions  and  markings,  with  the  stated  modifica- 
tions in  certain  cases,  will  be  carefully  followed  by 
deputy  surveyors  in  their  field  notes;  and  their  field  work 
will  strictly  comply  with  the  requirements  of  the 
descriptions. 

2.  When  pits  and  mounds  of  earth  are  made  accesso- 
ries to  corners,  the  pits  will  always  have  a  rectangular 
plan;  while  the  mounds  will  have  a  conical  form,  with 
circular  base;  and  in  all  cases  both  pits  and  mounds  will 
have  dimensions  at  least  as  great  as  those  specified  in  the 
descriptions.  Deputy  surveyors  will  strictly  adhere  to 


228  A    MANUAL    OF    LAND    SURVEYING. 

these  provisions,  and  no  departure  from  the  stated  re- 
quirements will  be  permitted,  either  in  instructions  or 
practice  in  tJie  field. 

3.  Kef  erring  to  the  numbered  paragraphs,  the  corners 
described  in  "3  "  will  be  preferred  to  those  described  in 
either  "1"  or  "2, J1  when  corners  are  established  in 
loose,  sandy  soil,  and  good  bearing  trees  are  available: 
under  similar  conditions,  the  corners  described  in  "5" 
and  "8"  will  be  preferred  to  those  described  in  "4" 
and  "  7,"  respectively. 

4.  The  selection  of  the  particular  construction  to  be 
adopted  in  any  case  will  be  left,  as  a  matter  of  course,  to 
the  judgment  and  discretion  of  the  deputy,  who  will  as- 
sign the  greatest  weight  to  the  durability  of  the  corner 
materials  and  pei-manency  of  the  finished  corners. 

Abbreviations  Allowed  in  Returns,  —  Dimen- 
sions of  stones,  posts,  and  pits  should  for  brevity 
be  expressed  in  a  regular  manner,  in  consecutive 
order  of  length,  breadth,  and  thickness,  as  shown  in 
specimens  ;  for  instance,  "  a  stone  23  x  10  x  8  ins." 
To  describe  a  mound  the  material,  the  altitude,  and 
diameter  of  base  will  be  given,  as  "mound  of  earth 
4  ft.  base,  2£  ft.  high." 

The  following  contractions  are  authorized  to  be 
used  in  the  preparation  of  field  notes,  transcripts, 
inspection  reports,  and  similar  records,  and  no  others 
should  be  introduced.  The  arrangement  of  lines, 
blanks,  spaces,  numbers,  and  the  general  form  of 
the  specimen  notes  should  be  observed. 


ORIGINAL    SURVEYS. 


229 


A. 

for  acres. 

M.  C. 

for  meander   cor- 

a. m. 

"  forenoon. 

ner. 

A.  M.  C. 

"  aux.  meander 

mer. 

1   meridian. 

corner. 

mkd. 

1   marked. 

asc. 

"  ascend. 

N. 

1  north. 

astron. 

'  astronomical. 

NE. 

'   northeast. 

bdy. 
bdrs. 

'   boundary. 
"  boundaries. 

NW. 
obs. 

1  northwest. 
"  observe. 

bet. 

"  between. 

obsn. 

"  observation. 

B.  0. 

"  bearing  ob- 
ject. 

p.  m. 
Pol. 

"  afternoon. 
"   Polaris. 

B.  T. 

"   bearing  tree. 

Pr.  Mer. 

"  principal    me- 

C. C. 

"  closing  cor- 

ridian. 

ner. 

Pt.  of  Tr. 

14   point  of  trian- 

chs. 

.  "   chains. 

gulation. 

cor.,  cors 

.  "   corner,  cor- 

Xsec. 

"  quarter  sec- 

ners. 

tion. 

corr. 

"  correction. 

R.,  Rs. 

"   range,  ranges. 

decl. 

"  declination 

red. 

"   reduce,  reduc- 

dep. 

"   departure. 

tion. 

desc. 

"  descend. 

S. 

;<  south. 

dia. 

"   diameter. 

S.  C. 

"   standard    cor- 

difT. 

'  difference. 

ner. 

(  1st. 

;<  distance. 

SE. 

"  southeast. 

D.  S. 

"   deputy  sur- 

sec., sees., 

"   section,   sec- 

veyor. 

tions. 

E. 

;<  east. 

S.  M.  C. 

"   special   mean- 

elong. 
frac. 

"  elongation. 
"  fractional. 

sq. 

der  corner. 
1    square. 

ft. 

"  foot.  feet. 

St.  Par. 

"   standard    par- 

G. M. 

11  guide  merid- 

allel. 

ian. 

SW. 

"  southwest. 

h.,  hrs., 

"  hour,  hours. 

T.,  or  Tp. 

1  township. 

ins. 

'  inches. 

Ts.,orTps 

'   townships. 

lat. 

"  latitude. 

temp. 

'  temporary. 

L.  C. 

"   lower  culmi- 

U. C. 

"  upper     culmi- 

nation. 

nation. 

Iks. 

"  links. 

var. 

'  variation. 

1.  m.  t. 

"   local  mean 

W. 

'  west. 

time. 

W.  C. 

'   witness    cor- 

long. 

1  longitude. 

ner. 

m. 

"  minutes. 

w.  corr. 

"  watch   correc- 

mag. 

"  magnetic. 

W.  P. 

tion. 
'   witness  point. 

w.  t. 

"  watch  time. 

230  A    MANUAL    OF    LAND    SURVEYING. 

* 
AUTHORIZED    FORMS    AND    DESCRIPTIONS  OF    CORNERS. 

The  forms  given  below  will  guide  the  surveyor  in 
the  choice  and  erection  of  monuments  and  acces- 
sories, and  the  same  forms  will  be  followed  in  pre- 
paring field  notes.  In  case  a  deputy  is  compelled  to 
choose  another  style  of  corner,  he  should  state  in 
his  notes  the  reasons  that  made  it  necessary  to  depart 
from  the  rules,  and  should  erect  a  monument  of 
equal  or  greater  permanence  than  the  one  prescribed. 

The  punctuation  marks  heretofore  shown  in 
former  editions,  to  be  used  with  letters  and  figures 
on  stones,  posts,  and  trees,  are  now  omitted,  for  the 
reason  that  they  are  neither  made,  nor  desired  to  be 
made,  in  the  actual  field  work,  and  hence  should  not 
be  inserted  in  the  official  returns. 

The  stated  dimensions  of  posts  are  minimum  ;  if 
posts  are  longer  than  3  feet,  the  extra  length  will  be 
placed  in  the  ground  ;  the  posts  will  in  no  case  pro- 
ject more  than  12  ins.  above  the  natural  surface  of 
the  earth. 

STANDARD  TOWNSHIP  CORNERS. 
METHOD  OF  MARKING. 

When  more  than  one  half  of  all  the  standard  town- 
ship and  section  corners  on  any  6  miles  of  a  base  line  or 
standard  parallel  are  stone  corners,  the  descriptions  in 
paragraphs  1  and  2,  if  the  corners  therein  described  are 
established,  will  be  modified  as  follows:  Strike  out  "S. 
C.,  on  N."  After  "marked,"  insert  the  words:  — 

"S.  C.,  13  N.  onN., 

22  E.  on  E.,  and 

21  E.  on  W.  faces;  " 

When  under  the  conditions  above  specified,  the  corner 
described  in  paragraph  1  is  established,  a  stake  may  be 
driven  in  the  east  pit  and  marked  instead  of  the  stone, 
and  described  as  exemplified  in  the  last  clause  of  para- 
graph 6. 


ORIGINAL    SURVEYS.  231 

1.  Stone,  with  P#.s  and  Mound  of  Earth. 

Set  a  —  stone,  —  x— X—  ins.,—  ins.  in  the  ground,  for 
standard  cor.  of  (e.  g.)  Tps.  13  TS.,  Rs.  21  and  22  E., 
marked  S.  C.  on  N.;  with  6  grooves  on  N.,  E.,  and  W. 
faces;  dug  pits,  30x24x12  ins.,  crosswise  on  each  line,  E. 
and  W.,  4  ft.,  and  X.  of  stone,  8  ft.  dist.;  and  raised  a 
mound  of  earth,  5  ft.  base,  2i  ft.  high,  N.  of  cor.  The 
direction  of  the  mound,  from  the  corner,  will  be  stated 
wherever  a  mound  is  built. 

2.  Stone,  with  Mound  of  Stone. 

Set  a  —  stone,  — X— X—  ins.,  —  ins.  in  the  ground,  for 
standard  cor.  of  (e.  g)  Tps.  13  N.,  Rs.  21  and  22  E.,  marked 
S.  C.,  on  X.;  with  6  grooves  on  X.,  E.,  and  W.  faces;  and 
raised  a  mound  of  stone,  2  ft.  base,  li  ft.  high,  X.  of 
cor.  Pits  impracticable.  Mound  of  stone  will  consist 
of  not  less  than  four  stones,  and  will  be  at  kast  H  ft. 
high,,  with  2  ft.  base. 

3.  Stone,  with  Bearing  Trees. 

Seta  — stone,  — X— X—  ins.,  —  ins.  in  the  ground, 
for  standard  cor.  of  (e.  g.)  Tps.  13  N.,  Rs.  21  and  22  E., 
marked  S.  C..  on  N.;  with  6  grooves  on  N.,  E.,  and  W. 
faces;  from  which 

A  — ,  —  ins.  diam.,  bears  N.  —  °  E.,  —  Iks.  dist., 
marked 

T.  13  X..  R.  22  E..  S.  31,  B.  T. 

A  — ,  —  ins.  diam.,  bears  X.  — °  W.,  —  Iks.  dist., 
marked 

T.  13  X.,  R,  21  E.,  S.  36,  B.  T. 

All  bearing  trees,  except  those  referring  to  quarter 
section  corners,  will  be  marked  with  the  townshi}),  range, 
md  section  in  which  they  stand. 

4.  Post,  with  Pits  and  Mound  of  Earth. 

Set  a  —  post,  3  ft.  long,  4  ins.  sq.,  with  marked  stone 
(charred  stake  or  quart  of  charcoal),  24  ins.  in  the 
ground,  for  standard  cor.  of  (e.  g.)  Tps.  13  X.,  Rs.  22  and 
23  E.,  marked 

S.  C.,  T.  13  N.  on  N. 

R.  23  E.,  S.  31  on  E.,  and 

R.  22  E.,  S.  36  on  W.  faces;  with  5  grooves  on  K,  E., 
and  W.  faces;  dug  pits,  30x24x12  ins.,  crosswise  on  each 


232  A    MANUAL    OF    LAND    SURVEYING. 

line,  E.  and  W.,  4  ft.,  and  N.  of  post,  8  ft.  dist.;  and 
raised  a  mound  of  earth,  5  ft.  base,  2i  ft.  high,  N.  of  cor. 

5.  Post,  with  Bearing  Trees. 

Set  a  —  post,  3  ft.  long,  4  ins.  sq.,  24  ins.  in  the  ground, 
for  standard  cor.  of  (e.  g.)  Tps.  18  N.,  Rs.  22  and  23 JE., 
marked 

S.  C.,  T.  13  N.  on  N., 

R.  23  E.  S.  31  on  E.,  and 

R.  22  E.,  S.  36  on  W.  faces:  with  6  grooves  on  N.,-EM 
and  W.  faces,  from  which 

A  — .  —  ins.  diam.,  bears  N.  -  °  E.,  —  Iks.  dist., 
marked 

T.  13  N.,  R.  23  E.,  S.  31,  B.  T. 

A  — ,  —  ins.  diam.,  bears  N.  — °  W.,  —  Iks.  dist., 
marked 

T.  13  N.,  R.  22  E.,  S.  36,  B.  T. 

6.  Mound  of  Earth,  with  Deposit,  and  Stake  in  Pit. 
Deposited  a  marked  stone  (charred  stake  or  quart  of 

charcoal)  12  ins.  in  the  ground,  for  standard  cor.  of  (e.  g.} 
Tps.  13  N.,  Rs.  22  and  23  E.;  dug  pits,  30X24X12  ins., 
crosswise  on  each  line,  N.,  E.,  and  W.  of  cor.,  5  ft.  dist.; 
and  raised  a  mound  of  earth,  5  ft.  base,  2i  ft.  high,  over 
deposit. 

In  E.  pit  drove  a  —  stake,  2  ft.  long,  2  ins.  sq.,  12  ins. 
in  the  ground  marked 

S.  C.,T.13N.  onN., 

R.  23  E.,  S.  31  onE.,  and 

R.  22  E.,  S.  36  on  W.  faces;  with  6  grooves  on  N.,  E., 
and  W.  faces. 
7  Tree  Corner,  with  Pits  and  Mound  of  Earth. 

A  — ,  —  ins.  diam.,  for  standard  cor.  of  (e.  g.)  Tps.  13 
ST.,  Rs.  22  and  23  E.,  I  marked 

S.  C.,  T.  13  N.  on  N., 

R.  23  E.,  S.  31  on  E.,  and 

R.  22  E.,  S.  36  on  W.  sides;  with  6  notches  on  "N.,  E., 
and  W.  sides;  dug  pits,  24x18x12  ins.,  crosswise  on  each 
line,  N.,  E.,  and  W.  of  cor.,  5  ft.  dist.;  and  raised  a 
mound  of  earth  around  tree. 
8.  Tree  Corner,  with  Bearing  Trees. 
•  A  — ,  —  ins.  diam.,  for  standard  cor.  of  (e.  g.)  Tps.  13 


ORIGINAL    SUKVKYS.  '    233 

N.,  Rs.  22  and  23  E.,  I  marked 

S.  C.,  T.  13  N.  on  N., 

R.  23E.,  S.  31  on  E.,  and 

R,  22  E.,  S.  36  on  W.  sides;  witli  6  notches  on  N.,  E., 
and  W.  sides;  from  which 

.v  — ,  —  ins.  diam.,  bears  N.  — °  E.,  —  Iks.  dist., 
marked 

T.  13  N.,  R.  23  E.,  S.  31,  B.  T. 

A  — ,  —  ins.  diam.,  bears  N:  — °  W.,  —  Iks.  dist., 
marked 

T.  13  X.,  R.  22  E.,  S.  36,  B.  T. 

Witness  Corners. —  1.  When  the  true  point  for 
any  corner  described  in  these  instructions  falls  where 
prevailing  conditions  would  insure  its  destruction  by 
natural  causes,  a  witness  corner  will  be  established  in  a 
secure  position,  on  a  surveyed  line  if  possible,  and  within 
twenty  chains  of  the  corner  point  thus  witnessed. 
2.  Markings  on  Witness  Corners. 

A  witness  corner  will  bear  the  same  marks  that  would 
be  placed  upon  the  corner  for  which  it  is  a  witness,  and 
in  addition,  will  have  the  letters  "  W.  C."  (for  witness 
corner),  conspicuously  displayed  above  the  regular  mark- 
ings; such  witness  corners  will  be  established,  in  all  other 
respects,  like  a  regular  corner,  marking  bearing  trees 
with  the  proper  numbers  for  the  sections  in  which 
they  stand. 

When  bearing  trees  are  described  as  accessories  to  a 
witness  corner,  the  prescribed  markings  on  each  tree 
will  be  preceded  by  the  letters  "W.  C.,"  distinctly  cut 
into  the  wood. 

The  true  bearing  and  distance  of  witness  corners, 
from  the  true  point  for  the  corner,  will  always  be  clearly 
stated  i  n  the  field  notes. 
4.   Witness  Comers  to  Corner  Points  Falling  in  Roads,  etc. 

The  point  for  a  corner  falling  on  a  railroad,  street,  or 
wagon  road,  will  be  perpetuated  by  a  marked  stone 
charred  stake  or  quart  of  charcoal,  deposited  24  inches 
in  the  ground,  and  witnessed  by  two  witness  corners,  one  of 
which  will  be  established  on  each  limiting  line  of  the 
highway. 


234  A    MANUAL    OF    LAND    SURVEYING. 

In  case  the  point  for  any  regular  corner  falls  at  the 
intersection  of  two  or  more  streets  or  roads,  it  will  be 
perpetuated  by  a  marked  stone  (charred  stake  or  quart 
of  charcoal),  deposited  24  inches  in  the  ground,  and 
ivitnessed  by  two  witness  corners  established  on 
opposite  sides  of  the  corner  point,  and  at  the  mutual 
intersections  of  the  lines  limiting  the  roads  or  streets, 
as  the  case  may  be. 

Witness  Points  will  be  perpetuated  by  corners 
similar  to  those  described  for  quarter  section  corners, 
with  the  marking  '•  W.  P."  (for  witness  point),  in  place 
of  "i,"or  "is. ", 'as  the  case  may  be. 

If  bearing  trees  are  available  as  accessories  to  witness 
points,  each  tree  will  be  marked  W.  P.  B.  T.  (See  "In- 
superable objects  on  line  —  Witness  Points." 

Miscellaneous. —  1.  Corners  on  Rock  in  Place,  or  on 
Boulders. 

When  a  corner  falls  on  rock  in  place,  or  on  a  boulder,  a 
cross  (X),  will  be  made  at  the  exact  corner  point,  and 
witnessed  by  the  proper  number  of  bearing  trees,  if  they 
are  available;  in  the  absence  of  suitable  trees,  a  mound 
of  stones  will  be  raised,  or  of  earth  if  stones  are 
not  found  and  pits  are  available.  Owing  to  the  diffi- 
culty of  identifying  the  corner  coming  upon  a  flat 
rock  in  place,  when  only  a  cross  is  cut  thereon,  it  is 
imperative  that  some  adequate  witness  be  used  and 
marked. 

2.  Location  of  Mounds. 

When  mounds  of  earth  or  other  material  are  raised 
as  accessories  to  corners,  they  will  be  placed  as  specified 
in  the  foregoing  Description  of  Corners,  and  in  every 
case  the  direction  of  tlie  mound  from  the  corner  will  be 
carefully  stated.  The  use  of  the  indefinite  description 
"alongside"  will  not  be  approved. 

In  case  the  character  of  the  land  is  such  that  the 
mound  cannot  be  placed  as  hereinbefore  described,  the 
deputy  will  state  in  his  notes,  by  bearing  and  distance, 
exactly  where  the  mound  is  located  with  reference  to 
the  corner,  and  will  give  his  reasons  for  placing  it  as 
described. 

3.  Mounds  of  Stone,  Covered  with  Earth. 


ORIGINAL    SURVEYS.  235 

In  a  case  where  pit8  are  practicable  and  the  deputy 
prefers  raising  a  mound  of  stone,  or  a  mound  of  stone 
covered  with  earth,  he  will  use  the  form  given  for 
"  Stone  with  mound  of  stone,"  when  the  corner  thus 
described  is  established;  but  when  the  corner  "Stone, 
with  mound  of  stone  covered  with  earth"  is  con- 
structed, the  description  will  be  modified  as  follows: 
Strike  out  the  words  "Pits  impracticable;"  in 
place  of  "  Mound  of  Stone,  2  ft.  base,  l£  ft.  high," 
write  "Mound  of  stone  covered  with  earth,  — ft. 
base,  —  ft.  high,"  inserting  in  the  blank  spaces 
the  dimensions  of  the  mound  given  in  paragraph  1, 
following  the  -designation  of  each  class  of  corners. 
Mounds  of  stone,  or  of  stone  covered  with  earth 
must  never  be  built  AROUND  the  corner  stone,  but 
separate.  When  stones  are  necessary  to  hold  the 
corner  stone  upright  and  firm,  they  should  be  in  ad- 
dition to  the  witness  mound,  and  not  a  part  of  it. 
4.  Bearing  Trees. 

Bearing  trees  marked  as  accessories  to  standard  cor- 
ners, either  township,  section,  or  quarter  section,  will 
be  selected  on  the  north  side  of  base  lines  or  standard 
parallels,  and  bearing  trees  referring  to  the  closing  cor- 
ners on  said  lines,  will  be  located  on  the  south  side;  in 
general,  the  bearing  trees  referring  to  any  particular 
closing  corner,  together  with  one  pit  and  the  mound  be- 
longing to  such  corner,  will  be  located  on  the  same  side  of 
tlie  line  dosed  upon,  and  on  the  side  from  which  the  surveys 
have  been  closed. 

When  the  requisite  number  of  trees  can  be  found 
within  300  links  of  the  corner  point,  two  (2)  bearing 
trees  will  be  marked  and  described  for  every  standard 
or  closing  township  or  section  corner,  or  corner  common 
to  two  townships  or  sections,  only;  four  (4)  for  every 
corner  common  to  four  townships  or  four  sections;  one 
(1)  for  a  corner  referring  to  one  township  or  one  section, 
only;  two  (2)  for  every  quarter  section  corner  or  meander 
corner,  and  four  (4)  for  each  mile  or  half  mile  corner,  or 
corner  monument  on  a  reservation  or  other  boundary, 
not  conforming  to  the  system  of  rectangular  surveying. 


236  A    MANUAL    OF    LAND    SURVEYING. 

The  limit  of  300  links  will  not  be  held  to  prohibit 
the  use  of  bearing  trees  or  rocks  beyond  that  dis- 
tance. Where  such  objects  are  few  but  accessible, 
they  are  too  useful  as  evidences  of  corners  to  be  dis- 
regarded by  a  faithful  deputy,  even  when  several 
chains  distant.  In  the  surveys  of  50  or  60  years 
ago,  corners  were  often  witnessed  by  trees  8  or  10 
chains  distant,  with  great  advantage  to  subsequent 
retracements. 

In  case  the  prescribed  number  of  trees  cannot  be 
found  within  practicable  distance,  the  deputy  will 
state  in  his  field  notes,  after  describing  those 
marked,  "No  other  trees  within  limits,"  and  add 
"Dig  pits — X — X — ins.,"  etc.,  or  "  Raise  a 
mound  of  stone,  — ft.  base,  — ft.  high,  — of  cor.," 
as  prevailing  conditions  may  require. 

Bearing  trees,  being  important  accessories  to 
the  corners,  will  have  their  exact  bearings  from  the 
true  meridian  taken  with  the  instrument  used  in  run- 
ning the  lines  of  survey  ;  and  the  distance  from  the 
middle  of  each  bearing  tree  to  the  middle  point  of  the 
corner  will  be  carefully  measured,  and  recorded  in 
the  field  notes. 

7.  As  to  the  height  or  position  of  marks  placed 
on  bearing  trees,  practice  differs  in  various  localities. 
The  custom  of  placing  these  important  evidences 
high  enough  to  insure  their  destruction  when  some 
woodman,  ignorant  or  careless  of  the  penalty  of  the 
law,  cuts  down  the  tree,  is  a  direct  violation  of 
rules.  A  tree  will  be  so  marked  that  if  inadvert- 
ently cut  down  its  stump  will  retain  evidence  of  its 
importance.  Many  surveyors  have  adopted  the 
plan  of  placing  all  the  marks  at  the  height  of  4  or  5 
feet,  except  the  letters  B  T,  which  are  made  on 
another  blaze  about  one  foot  above  the  ground. 
The  intent  is  commendable  ;  but  as  a  better  rule, 
applicable  to  trees  of  every  size,  the  following  is 
now  adopted  :  Place  all  figures  and  letters  on  that 
part  of  the  tree  which  would  probably  remain  as  the 
stump  ;  and  make  one  plain  blaze  high  on  the  same 


ORIGINAL    SURVEYS.  237 

side,  to  attract  notice  in  case  of  snow  or  dense 
undergrowth.  No  tree  less  than  4  inches  in  diameter 
should  be  chosen  for  a  witness,  if  larger  ones  are 
convenient:  and  if  none  over  3  inches  are  found, 
pits  will  be  dug  to  witness  the  corner. 

6.  Stones  for  Corners. 

Stones  18  ins.  long,  or  less,  will  be  set  with  two  thirds 
of  their  length  in  the  ground,  and  those  more  than  18 
ins.  long  will  have  three  fourths  of  their  length  in  the 
ground. 

~No  stones  measuring  less  than  504  cubic  inches,  or  less 
than  12  ins.  in  length,  will  be  used  for  corners. 

7.  Lines  Discontinued  at  Legal  Corners. 

No  mountainous  lands,  or  lands  not  classed  as  survey- 
able,  will  be  meandered,  and  all  lines  approaching  such 
lands  will  be  discontinued  at  the  section  or  quarter-sec- 
tion corner  nearest  the  unsurveyed  land. 

8.  Marks  to  be  cut. 

All  letters  and  figures  on  posts,  trees,  or  stones,  etc., 
will  be  cut  into  the  object  upon  which  they  are  placed. 
Arabic  figures  and  plain  letters  will  be  used  for  all 
markings. 

9.  Orientation  of  Corners. 

Corners  referring  to  one,  two,  or  four  townships  or 
sections,  not  identical  with  standard  or  closing  corners, 
will  be  set  with  their  faces  directed  XE.  and  SW.,  and 
N"W.  and  SE.,  while  all  other  corners  will  be  set  with 
their  sides  facing  the  cardinal  points;  except  corners  on 
boundaries  of  reservations  and  private  land  claims, 
which  will  be  set  squarely  on  line. 

10.  Size  of  Posts,  Hounds,  etc. 

The  sizes  of  wooden  posts,  mounds,  and  pits,  noted  in 
the  foregoing  descriptions,  will  be  regarded  as  minimum, 
and  their  dimensions  will  be  increased  whenever  prac- 
ticable. 

11.  Corner  Materials. 

In  establishing  corners  the  first  preference  will 
be  given  durable  stones  when  obtainable ;  then, 
posts  ;  and  lastly,  mounds,  with  stake  in  pit. 


238  A    MANUAL    OF    LAND    SURVEYING. 

Wood  of  a  perishable  nature  will  not  be  used  for 
posts  or  stakes. 
12.     Instructions  to  be  studied. 

Deputy  surveyors  will  carefully  read,  study,  and  fa- 
miliarize themselves  with  all  instructions  contained  in 
this  volume,  and  will  instruct  their  assistants  as  to 
their  duties  before  commencing  work.  An  extra  copy 
of  this  Manual  may  be  furnished  each  deputy,  for  the 
use  of  his  assistants. 

Initial  Points. —  Initial  points  from  which  the  lines 
of  the  public  surveys  are  to  be  extended  will  be  estab- 
lished whenever  necessary,  under  such  special  instruc- 
tions as  may  be  prescribed  in  each  case  by  the  Commis- 
sioner of  the  General  Land  Office.  The  locus  of  such 
initial  points  will  be  selected  with  great  care  and  due 
consideration  for  their  prominence  and  easy  identifica- 
tion, and  must  be  established  astronomically. 

An  initial  point  should  have  a  conspicuous  loca- 
tion, visible  from  distant  points  on  lines  ;  it  should 
be  perpetuated  by  an  indestructible  monument  pref- 
erably a  copper  bolt  firmly  set  in  a  rock  ledge  ;  and 
it  should  be  witnessed  by  rock  bearings,  without 
relying  on  anything  perishable  like  wood. 

115.  The  initial  point  having  been  established 
the  lines  of  public-land  surveys  will  be  extended 
therefrom.  They  are  classified  as  follows: 

Class  1.   Base  lines  and  standard  parallels. 

Class  2.   Principal  and  guide  meridians. 

Class  3.  Township  exteriors  (or  meridional  and 
latitudinal  township  boundaries). 

Class  4.   Subdivision  and  meander  lines. 

Only  the  base  line  and  principal  meridian  can 
pass  through  the  initial  point. 

Base  Line. — 1.  From  the  initial  point  the  base  line 
will  be  extended  east  and  west  on  a  parallel  of  latitude, 
by  the  use  of  transit  or  solar  instruments,  as  may  be 


ORIGINAL    SURVEYS.  269 

directed  by  the  surveyor  general  in  his  written  special 
instructions.  The  transit  will  be  used  for  the  aline- 
ment  of  all  important  lines. 

2.  The   direction   of   base    lines    will  conform  to 
parallels  of  latitude  and  will  be  controlled  by  true 
meridians;  consequently  the  correct  determination  of 
true  meridians  by  observations  on  Polaris  at  Elonga- 
tion is  a  matter  of  prime  importance. 

3.  Certain  reference  lines,  called  tangents  and  *e- 
cants,  having  a   known  position  and  relation  to  the 
required  parallel   of  latitude,  will   be  prolonged   as 
straight  lines.     Two  back  and  two  fore  sights  are 
taken  at  each  setting  of  the  instrument,   the  hori 
zontal  limb  being  revolved  180°  in  azimuth  between 
the  observations,  in  one  method,  taking  the  mean  of 
observations.     Another  method,  called  double  back 
and  fore  sights,  is  still  more  exact  and  therefore  pref- 
erable.    In    this    process  the  vertical  cross-wire  is 
fixed  upon  two  transit  points  at  some  distance  apart, 
in    the  rear,    and  then   reversed  to  set  one  or  two 
new    points   in    advance.     This    not    only  insures  a 
straight  line,  if  the  transit  is  leveled,  but  also  de- 
tects the  least  error  of  collimation. 

4.  Where   solar   apparatus  is  used  in  connection 
with  a  transit,  the  deputy  will  test  the  instrument, 
whenever  practicable,  by  comparing  its   indications 
with  a  meridian  determined  by  Polaris  observations; 
and  in  all  cases  where  error  is  discovered,  he  will 
make  the  necessary  corrections  of  his  line  before  pro- 
ceeding  with   the   survey.     All   operations  will  be 
fully  described  in  the  field  notes. 

5.  The  proper  township,  section,  and  quarter  section 
corners  will  be  established  at  lawful  intervals,  and  me- 
ander corners  at  the  intersection  of  the  line  with  all 
meanderable  streams,  lakes,  or  bayous. 

6.  In  order  to  detect  errors  and  insure  accuracy  in 
measurement,  two  sets  of  chainmen  will  be  employed; 
one  to  note  distances  to  intermediate  points  and  to  lo 


240  A    MANUAL    OF    LAND    SURVEYING. 

cate  topographical  features,  the  other  to  act  as  a 
check.  Each  will  measure  forty  chains  and  in  case 
the  difference  is  inconsiderable,  the  proper  corner 
will  be  placed  midway  between  the  ending  points  of 
the  two  measurements;  but  if  the  discrepancy  exceed 
8  links  on  even  ground.,  or  25  links  on  mountainous 
surface,  the  true  distance  will  be  found  by  careful 
re-chaining  by  one  party  or  both. 

The  deputy  will  be  present  when  each  corner  is  thup 
established,  and  will  record  in  the  body  of  his  field  notes 
the  distances  to  the  same,  according  to  the  measure- 
ment by  each  set  of  chainmen. 

To  obviate  collusion  between  the  sets  of  chainmen, 
the  second  set  should  commence  at  a  point  in  advance 
of  the  beginning  corner  of  the  first  set,  the  initial  dif- 
ference in  measurement  thus  obtained  being  known 
only  to  the  deputy 

Principal  Meridian.— 1.  This  line  shall  conform  to 
a  true  meridian  and  will  be  extended  from  the  initial 
point,  either  north  or  south,  or  in  both  directions,  as 
the  conditions  may  require,  by  the  use  of  transit  or  solar 
instruments,  as  may  be  directed  by  the  surveyor  general 
in  his  special  written  instructions. 

2.  The  methods  used  for  determination  of  directions, 
and  the  precautions  to-be  observed  to  secure  accuracy 
in  measurement,  are  fully  stated  above  under  the  title 
"Base  Line,"  and  will  be  complied  with  in  every  partic- 
ular. 

3.  In  addition  to  the  above  general  instructions,  it  is 
required  that  in  all  cases  where  the  establishment  of  a 
new  principal  meridian  seems  to  be  necessary  to  the 
surveyor  general,  he  shall  submit  the  matter,  together 
with  his  reasons  therefor,  to  the  commissioner  of  the 
General  Land  Office,  and  the  survey  of  such  principal 
meridian  shall  not  be  commenced  until  written  author- 
ity, together  with  such  special  instructions  as  he  may 
deem  necessary,  shall  have  been  received  from  the  com- 
missioner. 

Standard  Parallels.  — 1.  Standard  parallels,  which 
are  also  called  correction  lines,  shall  be  extended- east 


ORIGINAL    SURVEYS.  241 

and  west  from  the  principal  meridian,  at  intervals  of 
24  miles  north  and  south  of  the  base  line,  in  the 
manner  prescri  bed  for  running  said  line,  and  all  require- 
ments under  the  title  "Base  Line"  will  be  carefully 
observed. 

2.  Where  standard  parallels  have  been  placed  at  inter- 
vals of  30  or  36  miles,  regardless  of  existing  instructions, 
and  where  gross  irregularities  require  additional  stand- 
ard* lines,  from  which  to  initiate  new,  or  upon  which 
to  close  old  surveys,  an  intermediate  correction  line 
should  be  established  to  which  a  local  name  may  bi 
given,  e.  #.,  "Cedar  Creek  Correction  Line;"  and  the  . 
same  will  be  run,  in  all  respects,  like  the  regular  stan- 
dard parallels. 

Guide  Meridians-— 1.  Guide  meridians  shall  be 
extended  north  from  the  base  line,  or  standard 
parallels,  at  intervals  of  24  miles  east  and  west  from 
the  principal  meridian,  in  the  manner  prescribed  for 
running  the  principal  meridian,  and  all  the  provisions 
for  securing  accuracy  of  alinement  and  measurement 
found,  or  referred  to  under  the  titles  "Base  Line," 
and  "Principal  Meridian,"  will  apply  to  the  survey 
of  said  guide  meridians, 

2.  When  existing  conditions  require  that  such  guide 
meridians  shall  be  run  south  from  the  base  or  correction 
lines,  they  will  be  initiated  at  properly  established 
closing   corners   on    such   lines   marked   as   closing 
corners. 

3.  Where   guide   meridians   have   been   improperly 
placed  at  intervals  greatly  exceeding  the  authorized 
distance  of  24  miles,  and  standard  lines  are  required  to 
limit  errors  of  old,  or  govern  new  surveys,  a  new  guide 
meridian  may  be  run  from  a  standard,  or  properly  estab- 
lished closing  corner,  and  a  local  name  may  be  assigned 
to  the  same,  e.  </.,  "Grass  Valley  Guide  Meridian."' 
These  additional  guide  meridians  will  be  surveyed  in  all 
respects  like  the  regular  guide  meridians. 

Township  Exteriors.— 1.  Whenever  practicable, 
the  township  exteriors  in  a  block  of  laud  24  miles 
square,  bounded  by  standard  lines,  will  be  surveyed 
successively  through  the  block,  beginning  with  those 
of  the  southwestern  township. 


242  A    MANUAL    OF    LAND    SURVEYING. 

2.  The  meridional  boundaries  of  townships  will  have 
precedence  in  the  order  of  survey  and  will  be  run  from 
south  to  north  on  true  meridians,  with  permanent  corners 
at  lawful  distances;  the  latitudinal  boundaries  will  be 
run  from  east  to  west  on  random  or  trial  lines,  and  cor- 
rected back  on  true  lines. 

The  falling  of  a  random,  north  or  south  of  the  town- 
ship corner  to  be  closed  upon,  will  be  carefully  measT 
ured,  and,  with  the  resulting  true  return  course,  will  be 
duly  recorded  in  the  field  notes. 

Should  it  happen,  however,  that  such  random  inter- 
sects the  meridian  of  the  objective  corner,  north  or 
south  of  said  corner,  or  falls  short  of,  or  overruns  the 
length  of  the  south  boundary  of  the  township  by  more 
than  three  chains  (due  allowance  being  made  for  conver- 
gency),  said  random,  and,  if  necessary,  all  the  exterior 
boundaries  of  the  township,  will  be  retraced  and  re- 
measured  to  discover  and  correct  the  error 

When  running  random  lines  from  east  to  west,  tem- 
porary corners  will  be  set  at  intervals  of  40.00  chains, 
and  proper  permanent  corners  will  be  established  upon 
the  true  line,  corrected  back  in  accordance  with  these 
instructions,  thereby  throwing  the  excess  or  deficiency 
against  the  west  boundary  of  the  township,  as  required 
by  law. 

3.  Whenever  practicable,  the  exterior  boundaries  of 
townships  belonging  to  the  west  range,  in  a  tract  or 
block  24  miles  square,  will  first  be  surveyed  in  succes- 
sion, through  the  range,  from  south  to  north;  and  in  a 
similar  manner,  the  other  three  ranges  will  be  surveyed 
in  regular  sequence. 

4.  In  cases  where  impassable  objects  occur  and  the  forego- 
ing rules  cannot  be  complied  with,  towrnship  corners  will 
be  established  as  follows:  — 

In  extending  the  south  or  north  boundaries  of  a  town- 
ship to  the  west,  where  the  southwest  or  northwest  corners 
cannot  be  established  in  the  regular  way  by  running  a 
north  and  south  line,  such  boundaries  will  be  run  west 
on  a  true  line,  allowing  for  convergency  on  the  west  half 


ORIGINAL    SURVEYS.  "  243 

mile;  and  from  the  township  corner  established  at  the 
end  of  such  boundary,  the  west  boundary  will  be  run 
north  or  south,  as  the  case  may  be.  In  extending  south  or 
north  boundaries  of  a  township  to  the  east,  where  the 
southeast  or  northeast  corner  cannot  be  established  in  the 
regular  way,  the  same  rule  will  be  observed,  except  that 
such  boundaries  will  be  run  east  on  a  ti-ue  line,  and  the 
east  boundary  run  north  or  south,  as  the  case  may  be. 

5.  Allowance  for  the  convergency  of  meridians  will  be 
made  whenever  necessary. 

Method  of  Subdividing.— 1.  The  exterior  bound- 
aries of  a  full  township  having  been  properly  estab- 
lished so  far  as  possible  the  subdivision  thereof  will 
be  made  as  follows: — 

At  or  near  the  southeast  corner  of  the  township,  a 
true  meridian  will  be  determined  by  Polaris  or  solar 
observations,  and  the  deputy's  instrument  will  be 
tested  thereon;  then  from  said  corner  the  first  mile  of 
the  east  and  south  boundaries  will  be  retraced,  if 
subdivisions  and  survey  of  the  exteriors  have  be"eu 
provided  for  in  separate  contracts;  but  if  the  survey 
of  the  exterior  and  subdivisional  lines  are  included 
in  the  same  contract,  the  retracements  referred  to  will 
be  omitted.  All  discrepancies  resulting  from  disagree- 
ment of  bearings  or  measurements  will  be  carefully 
stated  in  the  field  notes. 

The  meridional  sectional  lines  will  be  made 
parallel  to  the  range  line  or  east  boundary  of  the 
township,  by  applying  to  the  bearing  of  the  latter  a 
small  correction  dependent  on  the  latitude,  taken 
from  the  following  table,  which  gives,  to  the  nearest 
whole  minute,  the  convergency  of  two  meridians  6 
miles  long  and  from  1  to  5  miles  apart ;  and  sup- 
plies directly  the  deviation  of  meridional  section 
lines  west  of  north,  when  the  range  line  is  a  true 
meridian.  Add  the  correction  to  the  bearing  of  the 
range  line,  if  the  same  is  west  of  north,  but  subtract 
when  it  bears  east  of  north. 


'244 


A    MANUAL    OF    LAND    SURVEYING. 


TABLE  II. — Corrections  for  Convergency  within  a 
Towtiship. 


Latitude. 

Correction  to  be  applied  to  bearing  of 
range  lines  at  a  distance  of  — 

1  mile. 

2  miles. 

3  miles. 

4  miles. 

5  miles. 

0             0 

/ 

/ 

/ 

I 

' 

30  to  35..   .. 

1 

1 

2 

2 

3 

35  to  40..   .. 

1 

1 

2 

3 

3 

40  to  45..   .. 

1 

2 

2 

3 

4 

45  to  50..   . 

1 

2 

3 

4 

5 

50  to  55..   . 

1 

2 

3 

5 

6 

55  to  60..   . 

1 

3 

4 

5 

7 

60  to  65..   . 

2 

3 

5 

7 

8 

65  to  70..   . 

2 

4 

6 

8 

10 

w. 
w. 
w. 


Example. — Latitude  47°.  Range  line  bears  N. 
0°  2'  E. ;  then  parallel  meridional  section  lines  will 
be  run  as  follows: 

From  the  corner  for  sections — 
35  and  36,  K.  0°  l'  E. 
34  and  35,  north. 
33  and  34,  N.  0°  1 
32  and  33,  N.  0°  2 
31  and  32,  N.  0°  3 

2.  After  testing  his  instrument  on  the  true  meridian 
thus  determined,  the  deputy  will  commence  at  the  cor- 
ner to  sections  35  and  36,  on  the  south  boundary,  and 
run  a  line  paralkl  to  the  range  line,  establishing  at  40.00 
chains,  the  quarter  section  corner  between  sections  35 
and  36,  and  at  80.00  chains  the  corner  for  sections  25,  26, 
35,  and  36. 

3.  From  the  last-named  corner,  a  random  line  will  be 
run  eastward,  without  blazing,  parallel  to  the  south  bound-- 
ary  of  section  36,  to  its  intersection  with  the  east  bound- 
ary of  the  township,  placing  at  40.00  chains  from  the 
point  of  beginning,  a  post  for  temporary  quarter  section 
corner.    If  the  random  line  intersects  said  township 


ORIGINAL    SURVEYS.  245 

boundary  exactly  at  the  corner  for  sections  25  and  36,  it 
will  be  blazed  back  and  established  as  the  true  line,  the 
permanent  quarter  section  corner  being  established 
thereon,  midway  between  the  initial  and  terminal  sec- 
tion corners. 

When  tne  objective  corner  is  in  sight  from  the 
starting  corner,  or  the  deputy  has  evidence  of  its 
location  to  prove  that  a  different  random  course 
would  fall  closer  to  the  corner,  he  may  use  such 
changed  course  for  his  random.  A  line  may  be  run 
as  a  "random  for  distance  only,"  when  the  course  is 
certain. 

If,  however,  the  random  intersects  said  township 
boundary  to  the  north  or  south  of  said  corner,  the  fall- 
ing will  be  carefully  measured,  and  from  the  data  thus 
obtained,  the  true  return  course  will  be  calculated,  and 
the  true  line  blazed  and  established,  and  the  position  of 
the  quarter  section  corner  determined,  as  directed 
above. 

The  details  of  the  entire  operation  will  be  recorded  in 
the  field  notes. 

4.  Having  thus  established  the  line  between  sections 
25  and  36;  from  the  corner  for  sections  25,  26,  35  and  36, 
the  west  and  north  boundaries  of  sections  25,  24,  13,  and 
12,  will  be  established  as  directed  for  those  of  section 
36;  with  the  exception  that  the  random  lines  of  said 
north  boundaries  will  be  run  parallel  to  the  established  south 
boundaries  of  the  sections  to  which  they  belonq,  instead  of  the 
south  boundary  of  section  36;  e.  g.  the  random  line  be- 
tween sections  24  and  25  will  be  run  parallel  to  the  es- 
tablished south  boundary  of  section  25,  etc. 

5.  Then,  from  the  last  established  section  corner,  i.  e. 
the  corner  for  sections  1,  2,  11,  and  12,  the  line  between 
sections  1  and  2,  will  be  projected  northward,  on  a 
random  line,  parallelto  the  east  boundary  of  the  town- 
ship, setting  a  post  for  temporary  quarter  section  cor- 
ner at  40.00  chains,  to  its  intersection  with  the  north 
boundary  of  the  township.     If  the  random  intersects 
said  north  boundary  exactly  at  corner  for  sections  1 
and  2,  it  will  be  blazed  back  and  established  as  the 


246  A    MANUAL    OF    LAND    SURVEYING. 

true  line,  the  temporary  quarter  section  comer  being 
established  permanently  in  its  original  position,  and 
the  fractional  measurement  thrown  into  that  portion 
of  the  line  between  said  corner  and  the  north  boundary 
of  the  township. 

If,  however,  said  random  intersects  the  north  bound- 
ary of  the  township,  to  the  east  or  west  of  the  corner 
for  sections  1  and  2,  the  consequent  falling  will  be  care- 
fully measured,  and  from  the  data  thus  obtained,  the 
true  return  course  will  be  calculated,  and  the  true  line 
established;  the  permanent  quarter  section  corner  being 
placed  upon  the  same  at  40.00  chains  from  the  initial 
corner  of  the  random  line,  thereby  throwing  the  frac- 
tional measurement  in  that  portion  lying  between  the 
quarter  section  corner  and  the  north  boundary  of  the 
township. 

When  the  north  boundary  of  a  township  is  a  base  line 
or  standard  parallel,  the  line  between  sections  1  and  2 
will  be  run  parallel  to  the  range  line  as  a  true  line,  the  quar- 
ter section  corner  will  be  placed  at  40.00  chains,  and  a 
closing  corner  will  be  established  at  the  point  of  inter- 
section with  such  base  or  standard  line ;  and  in  such 
case,  the  distance  from  said  closing  corner,  to  the  near- 
est standard  corner  on  such  base  or  standard  line, 
will  be  carefully  measured  and  noted  as  a  connection 
line. 

6.  Each  successive  range  of  sections  progressing  to 
the  west,  until  the  fifth  range  is  attained,  will  be  sur- 
veyed in  a  similar  manner;  then,  from  the  section  cor- 
ners established  on  the  west  boundary  of  said  range  of 
sections,  random  lines  will  be  projected  to  their  inter- 
section with  the  west  boundary  of  the  township,  and 
the  true  return  lines  established  as  prescribed  for  the 
survey  of  the  first  or  most  eastern  range  of  sections, 
with  the  exception  that  on  the  true  lines  thus  estab- 
lished, the  quarter-section  corners  will  be  established  at 
40. 00  chains  from  the  initial  corners  of  the  randoms,  the 
fractional  measurements  being  thereby  thrown  into 
those  portions  of  the  lines  situated  between  said  quar- 
ter-section corners  and  the  west  boundary  of  the  town- 
ship. 


ORIGINAL    SURVEYS.  247 

7.  The  following  general  requirements  are  reiterated 
for  emphasis:  — 

The  random  of  a  latitudinal  section  line  will  always 
be  run  parallel  to  the  south  boundary  of  the  section  to 
which  it  belongs,  and  with  the  true  bearing  of  said 
boundary;  and  when  a  section  has  no  linear  south 
boundary,  the  random  will  be  run  parallel  to  the  south 
boundary  of  the  range  of  sections  in  which  it  is  situ- 
ated, and  fractional  true  lines  will  be  run  in  a  similar 
manner. 

8.  The  deputy  is  not  required  to  complete  the  survey 
of  the  first  range  of  sections  from  south  to  north  before 
commencing  the  survey  of  the  second  or  any  subsequent 
range  of  sections,  but  the  corner  on  which  any  random 
line  closes  shall  have  been  previously  established  by 
running  the  line  which  determines  its  position,  except 
as  follows:  Where  it  is  impracticable  to  establish  such 
section  corner  in  the  regular  manner,  it  will  be  estab- 
lished by  running  the  latitudinal  section  line  as  a  true 
line,  with  a  true  bearing,  determined  as  above  directed 
for  random  lines,  setting  the  quarter-section  corner  at 
40.00  chains  and  the  section  corner  at  80.00  chains. 

9.  Quarter-section  corners,  both  upon  meridional  and 
latitudinal  section  lines,  will  be  established  at  points 
equidistant  from  the  corresponding  section  corners,  except 
upon  the  lines  closing  on  the  north  and  west  boundaries 
of  the  township,  and  in  those  situations  the  quarter- 
section  corners  will  always  be  established  at  precisely 
forty  chains  to  the  north  or  west  (as  the  case  may  be)  of 
the  respective  section  corners  from  which  those  lines 
respectively  start,  by  which  procedure  the  excess  or  de- 
ficiency in  the  measurements  will  be  thrown,  according 
to  law,  on  the  extreme  tier  or  range  of  quarter  sections, 
as  the  case  may  be. 

10.  Where  by  reason  of  impassable  objects  only  a  por- 
tion of  the  south  boundary  of  a  township  can  be  estab- 
lished, an  auxiliary  base  line  {or  lines,  as  the  case  may 
require)  will  be  run  through  the  portion  which  has  no 
linear  south  boundary,  first  random,  then  corrected, 
connecting  properly-established  corresponding  section 


A    MANUAL    OF    LAND    SURVEYING. 

corners  (either  interior  or  exterior)  and  as  far  south  as 
possible,  and  from  such  line  or  lines,  the  section  lines 
will  be  extended  northwardly  in  the  usual  manner,  and 
any  fraction  south  of  said  line  will  be  surveyed  in  the 
opposite  direction  from  the  section  corners  on  the  aux- 
iliary base  thus  established. 

11.  Where  by  reason  of  impassable  objects  no  portion 
of  the  south  boundary  of  a  township  can  be  regularly  estab- 
lished, the  subdivision  thereof  will  proceed  from  north 
to  south  and  from  east  to  west,  thereby  throwing  all  frac- 
tional measurements  and  areas  against  the  west  bound- 
ary, and  the  meanderable  stream  or  other  boundary 
limiting  the  township  on  the  south. 

If  the  east  boundary  is  without  regular  section  corners 
and  the  north  boundary  has  been  run  eastwardly  as  a 
true  line,  with  section  corners  at  regular  intervals  of 
80.00  chains,  the  subdivision  of  the  township  will  be 
made  from  west  to  east,  and  fractional  measurements  and 
areas  will  be  thrown  against  the  irregular  east  bound- 
ary. 

12.  When  the  proper  point  for  the  establishment  of  a 
township  or  section  corner  is  inaccessible,  and  a  witness 
corner  can  be  erected  upon  each  of  the  two  lines  which 
approach  the  same,  at  distances  not  exceeding  twenty 
chains  therefrom,  said  witness  corners  will  be  properly 
established,  and  the  half  miles  upon  which  they  stand 
will  be  recognized  as  surveyed  lines. 

The  witness  corner  will  be  marked  as  conspicuously 
as  a  section  corner,  and  bearing  trees  will  be  used  wher- 
ever possible. 

The  deputy  will  be  required  to  furnish  good  evidence 
that  the  section  corner  'is  actually  inaccessible. 

Where  impassable  precipices,  deep  canyons,  or 
lands  otherwise  quite  un  survey  able,  prevent  the  ex- 
tension of  regular  lines,  deputies  are  not  authorized 
to  set  meander  corners,  nor  to  meander  the  line  sep- 
arating lands  that  can  be  traversed  from  those  that 
cannot.  In  place  of  meandering,  they  are  to  set 


ORIGINAL    SURVEYS.  249 

witness  corners  on  line,  near  the  intersection  of  sec- 
tion lines  with  the  brink  or  foot  of  the  impassable 
cliffs,  or  at  the  margin  of  the  impracticable  m.irsh, 
to  represent  an  inaccessible  regular  section  or 
quarter  section  corner  if  within  twenty  chains. 
Such  quarter  sections  thus  marked  may  be  platted 
as  surveyed. 

Where  a  large  or  desirable  tract  is  found  to  have 
its  accessible  section  lines  too  short  to  justify  the 
erection  of  such  witness  corners,  and  to  render  it 
regularly  surveyed,  offset  lines  may  be  run  on  lines 
of  legal  subdivision,  far  enough  to  show,  by  neces- 
sary witness  corners,  the  40-acre  tracts  that  would 
otherwise  have  been  excluded  from  survey. 

The  topographic  sketches  of  mesas  and  impassable 
canyon  regions,  returned  by  deputies,  will  show  as 
nearly  as  practicable  the  location  of  these  features 
and  their  margins  ;  and  where  possible  the  corners 
on  opposite  sides  of  a  canyon  should  be  connected  by 
triangulation  at  least  once  in  each  township. 

Meandering. — i.  Lands  bounded  by  waters  are  to 
be  meandered  at  mean  high-watermark.  This  term 
has  been  defined  in  a  State  decision  (47  Iowa,  370) 
in  substance  as  follows:  High- water  mark  in  the 
Mississippi  River  is  to  be  determined  from  the  river 
bed ;  and  that  only  is  river  bed  which  the  river  oc- 
cupies long  enough  to  wrest  it  from  vegetation. 
In  cases  where  the  deputy  finds  it  impossible  to 
carry  his  meander  line  along  mean  high-water  mark, 
his  notes  should  state  the  distance  therefrom,  and 
the  obstacles  which  justify  the  deviation. 

Proceeding  down  stream,  the  bank  on  the  left 
hand  is  termed  the  left  bank  and  that  on  the  right 
hand  the  right  bank.  These  terms  will  be  univers- 
ally used  to  distinguish  the  two  banks  of  a  river  or 
stream. 

2.  Navigable  rivers,  as  well  as  all  rivers  not  embraced 
in  the  class  denominated  "navigable,"  the  right-angle 
width  of  which  is  three  chains  and  upward,  will  be 


250  A    MANUAL    OF    LAND    SURVEYING. 

meandered  on  both  banks,  at  the  ordinary  mean  high 
watermark,  by  taking  the  general  courses  and  distances 
of  their  sinuosities,  and  the  same  will  be  entered  in  the 
field  book.  Rivers  not  classed  as  navigable  will  not  be 
meandered  above  the  point  where  the  average  right- 
angle  width  is  less  than  three  chains,  except  that 
streams  which  are  less  than  three  chains  wide  and 
which  are  so  deep,  swift,  and  dangerous  as  to  be  im- 
passable through  the  agricultural  season,  may  be 
meandered,  where  good  agricultural  lands  along  the 
shores  require  their  separation  into  fractional  lots  for 
the  benefit  of  settlers.  But  such  meander  surveys 
shall  be  subject  to  rejection  if  proved  unnecessary 
by  field  inspection. 

Shallow  streams,  without  any  well-defined  chan- 
nel or  permanent  banks,  will  not  be  meandered ; 
except  tide-water  streams,  whether  more  or  less  than 
three  chains  wide,  which  should  be  meandered  at 
ordinary  high-water  mark,  as  far  as  tide-water 
extends. 

At  every  point  where  either  standard,  township, 
or  section  lines  intersect  the  bank  of  a  navigable 
stream,  or  any  meanderable  shore,  corners  will  be 
established  at  the  time  of  running  these  lines. 
Such  corners  are  called  meander  corners,  and  the 
deputy  will  commence  at  one  of  these  corners,  fol- 
low the  bank  or  boundary  line,  and  take  the  bearing 
and  measure  the  length  of  each  course,  from  the  be- 
ginning corner  to  the  next  meander  c^rnor. 

Regular  meander  corners  are  those  established  on 
standard,  township,  or  section  lines. 

The  meander  corners  on  lines  of  legal  subdivisions, 
other  than  standard,  township  or  section  lines,  will 
be  designated  special  meander  corners. 

Meander  corners,  not  on  a  line  belonging  to  the 
system  of  rectangular  surveying,  will  be  called 
auxiliary  meander  corners. 

When  a  Meander  Corner  falls  at  a  point  where 
prevailing  conditions  would  threaten  its  destruction 
by  natural  causes,  a  witness  corner  to  such  meander 


ORIGINAL    SURVEYS.  251 

corner  will  be  established,  as  provided  for  in  the 
article  Witness  Corners. 

All  courses  reported  are  to  be  compass  courses, 
taken  or  counted  from  the  meridian,  and  not  from  a 
latitudinal  line  ;  and  "transit  angles"  showing  only 
the  amount  of  deviation  from  the  preceding  course, 
are  not  allowed  in  field  notes  of  meanders. 

For  convenience  of  testing  by  traverse,  the  courses 
of  meander  lines  should  be  given  by  the  nearest 
quarter  degree.  '  As  meandered  lines  are  not  strict 
boundaries,  this  method  will  give  results  with  ap- 
proximate accuracy  for  good  closings  within  the 
limits  of  a  section.  Meander  lines  will  be  examined 
in  the  field  as  well  as  rectangular  lines,  before  ac- 
ceptance. 

All  meanders  should  be  traversed  before  leaving 
the  vicinity,  and  if  misclosure  is  found  indicating 
error  in  measurement  or  in  reading  courses,  the  linee 
must  be  re-meandered. 

The  crossing  distance  between  meander  corners  on 
same  line  and  the  true  bearing  and  distance  between 
corresponding  and  meander  corners,  will  be  ascer- 
tained by  triangulation  or  direct  measurement,  in 
order  that  both  shores  may  be  protracted.  The  par- 
ticulars will  be  given  in  the  field  notes. 

For  convenience  of  platting  and  computation,  the 
deputy  is  required  to  use  in  meanders  distances  hav- 
ing full  chains  or  multiples  of  ten  links,  with  odd 
links  only  in  closing  distances. 

3.  The  meanders  of  all  lakes,  navigable  bayous,  and 
deep  ponds,  of  the  area  of  twenty-five  acres  and  up- 
wards, will  be  commenced  at  a  meander  corner  and  con- 
tinued, as  above  directed  for  navigable  streams;  from 
said  corner,  the  courses  and  distances  of  the  entire  mar- 
gin of  the  same,  and  the  intersections  with  all  meander 
corners  established  thereon  will  be  noted. 

All  streams  falling  into  the  river,  lake,  or  bayou  will 
be  noted,  and  the  width  at  their  mouths  stated;  also, 
the  position,  size,  and  depth  of  springs,  whether  the 
water  be  pure  or  mineral;  also  the  heads  and  mouths  of 


252  A    MANUAL    OF    LAND    SURVEYING. 

all  bayous;  all  islands,  rapids,  and  bars  will  be  noted, 
with  Intersections,  to  their  upper  and  lower  ends,  to  es- 
tablish their  exact  situation.  The  elevation  of  the 
banks  of  lakes,  bayous,  and  streams,  the  height  of  falls 
and  cascades,  and  the  length  and  fall  of  rapids  will  be 
recorded  in  the  field  notes. 

To  meander  a  lake  or  deep  pond  lying  entirely  within 
the  boundaries  of  a  section,  two  lines  will  be  run  from 
the  two  nearest  corners  on  different  sides  of  such  lake 
or  pond,  the  courses  and  length  of  which  will  be  re- 
corded, and  if  coincident  with  unsurveyed  lines  of  legal 
subdivisions,  that  fact  will  also  be  stated  in  the  field 
notes,  and  at  each  of  the  points  where  said  lines  inter- 
sect the  margin  of  the  pond  or  lake,  a  special  meander 
corner  \yill  be  established  as  above  directed.  • 

The  relative  position  of  these  points  being  thus  defi- 
nitely fixed  in  the  section,  the  meandering  will  com- 
mence at  one  of  them  and  be  continued  to  the  other, 
noting  the  intersection,  and  thence  to  the  beginning. 
The  proceedings  are  to  be  fully  entered  in  the  field  notes. 

4.  Meander  lines  will  not  be  established  at  the  segre- 
gation line  between  dry  and  swamp  or  overflowed  land, 
but  at  the  ordinary  high-water  mark  of  the  actual  margin 
of  the  rivers  or  lakes  on  which  such  swamp  or  overflowed 
lands  border. 

5.  The  precise  relative  position  of  an  island,  in  a  town- 
ship made  fractional  by  a  river  or  lake  in  which  the  isl- 
and is  situated,  will  be  determined  by  triangulation 
from  a  special  and  carefully  measured  base  line,  initi- 
ated from  the  surveyed  lines,  on  or  near  the  lake  or 
river  bank  on  the  main  land,  so  as  to  connect  by  course 
and  distance  on  a  direct  line,  the  meander  corner  on  the 
mainland  with  the  corresponding  point  on  the  island, 
where  the  proper  meander  corner  will  be  established. 

6.  In  making  the  connection  of  an  island  lying  en- 
tirely within  a  section,  with  the  mainland,  a  special 
base  will  be  measured  from  the  most  convenient  mean- 
der corner,   and  from  such  base,  the  location  of  an 
auxiliary  meander  corner  will  be  determined  by  triangu- 
lation, at  which  the  meanders  of  the  island  will  be 
initiated. 

7.  In  the  survey  of  lands  bordering  on  tide  waters, 
meander  corners  may  be  temporarily  set  at  the  inter- 
section of  the  surveyed  lines  with  the  line  of  mean 
high  tide,  but  no  monument  should  be  placed  in 
a  position  exposed  to  the  beating  of  waves,  and 
the  action  of  ice  in  severe  weather.  In  all  such 
cases,  the  rule  given  in  section  90  must  be  observed 
by  establishing  a  witness  corner  on  line  at  a  secure 
point  near  the  true  point  for  the  meander  corner. 


ORIGINAL    SURVEYS.  'Jo3 

8.  The  field  notes  of  meanders  will  show  the  dates  on 
which  the  work  was  performed,  as  illustrated  in  the 
specimen  notes.  The  field  notes  of  meanders  will  state 
and  describe  the  corner  from  which  the  meanders 
commenced,  and  upon  which  they  closed,  and  will 
exhibit  the  meanders  of  each  fractional  section  sepa- 
rately; following,  and  composing  a  part  of  such  notes, 
will  be  given  a  description  of  the  land,  timber,  depth  of 
inundation"  to  which  the  bottom  is  subject,  and  the 
banks,  current,  and  bottom  of  the  stream  or  body  of 
water  meandered.  The  utmost  care  will  be  taken  to 
pass  no  object  of  topography,  or  change  therein,  without 
giving  a  particular  description  thereof  in  its  proper 
place  in  the  notes  of  the  meanders. 

Summary  of  Objects  and  Data  Required  to  be 
Noted. — 1.  The  precise  course  and  length  of  every 
line  run,  noting  all  necessary  offsets  therefrom,  with 
the  reason  for  making  them,  and  method  employed. 

2.  The  kind  and  diameter  of  all  bearing  trees,  with 
the  course  and  distance  of  the  same  from  their  respect- 
ive corners:  and  the  precise  relative  position  of  witness 
corners  to  the  true  corners. 

3.  The  kind  of  materials  of  which  corners  are  con- 
structed. 

4.  Trees  on  line.    The  name,  diameter,  and  distance  on 
line  to  all  trees  which  it  intersects. 

5.  Intersections   by    line   of   land  objects.       The 
distance  at  which  the  line  intersects  the  bound' try 
lines  of  every  reservation,  town  site,  duration  claim, 
Indian  allotment,   settler's  claim,   improvement,    or 
rancho;  prairie,  bottom  land,  swamp,  marsh,  grove, 
and  windfall,    with  the  course  of  the    same   at   all 
points  of  intersection;    also  the  distances  at  which 
the    line    begins    to    ascend,    arrives    at    the    top, 
begins    to    descend    and    reaches    the    foot   of   all 
remarkable    hills   and    ridges,    with   their   courses, 
and  estimated  height  in  feet,  above  the  level  land  of 
the  surrounding  country,  or  above  the  bottom  lands, 
ravines,    or   waters    near   which    they    are   situated. 
Also,  distance  to  and  across  large  ravines,  their  depth 
and  course. 

6.  Intersections  by  line  of  water  objects.    All  rivers, 
creeks,  and  smaller  ^streams  of  water  which  the  line 
crosses;  the  distances  measured  on  the  true  line  to  the 
bank  Jii-at  arrired  at,  the  course  dmm  stream  at  points  ot 
intersection,  and  their  widths  on  line.     In  cases  of 
navigable  streams,  their  width  will   be  ascertained 


254  A    MANUAL    OF    LAND    SURVEYING. 

between  the  meander  corners,  as  set  forth  under  the 
proper  head. 

7.  The    land's    surface — whether    level,  rolling, 
broken,  hilly,  or  mountainous. 

8.  The  soil — whether  rocky,  stony,   sandy,   clay, 
etc.,  and  also  whether  first,  second,  third,  or  fourth 
rate. 

9.  Timber — the  several  kinds  of  timber  and  under- 
growth, in  the  order  in  which  they  predominate. 

10.  Bottom  lands — to  be  described  as  wet  or  dry, 
and  if  subject  to  inundation,  state  to  what  depth. 

11.  Springs   of  water — whether  fresh,  saline,  or 
mineral,  with  the  course  of  the  stream  flowing  from 
them. 

12.  Lakes  and  ponds — describing  their  banks  and 
giving    their    height,    and   whether   it  be   pure   or 
stagnant,  deep  or  shallow. 

13.  Improvements.  —  Towns  and  villages  ;  houses 
or  cabins,  fields,  or  other  improvements  with  owner's 
names  ;  mill   sites,  forges   and  factories,  U.  S.  min- 
eral monuments,  and   all   corners   not    belonging  to 
the  system  of  rectangular  surveying  ;  will  be  located 
by  bearing  and  distance,  or  by  intersecting  bearings 
from  given  points. 

14.  Coal  banks   or  beds  ;  peat  or  turf    grounds  ; 
minerals  and  ores  with  particular  description  of  the 
same  as  to    quality  and  extent,   and    all    diggings 
therefor  ;  also  salt  springs  and  licks.     All  reliable 
information  that   can  be  obtained  respecting  these 
objects,   whether  they  be  on  the  line  or  not,  will 
appear  in  the  general  description. 

15.  Roods    and    trails,    with    their    directions, 
whence  and  whither. 

16.  Rapids,  cataracts,  cascades,  or  falls  of  water, 
with  the  estimated  height  of  their  fall  in  feet. 

17.  Precipices,    caves,    sink   holes,    ravines,    re- 
markable crags,  stone  quarries,  ledges  of  rocks,  with 
the  kind  of  stone  they  afford. 

18.  Natural  curiosities,   interesting  fossils,  petri- 
factions, organic  remains,  etc. ;  also  all  ancient  works 
of  art,  such  as  mounds,  fortifications,  embankments, 
ditches,  or  objects  of  like  nature. 

19.  The  magnetic  declination  will  be  incidentally 


ORIGINAL    SURVEYS.  255 

noted  at  all  points  of  the  lines  being  surveyed,  where 
any  material  change,  in  the  same  indicates  the  prob- 
able presence  of  iron  ores;  and  the  position  of  such 
points  will  be  perfectly  identified  in  the  field  notes. 

PRESCRIBED    LIMITS    FOR    CLOSINGS    AND    LENGTHS 
OF    LINES. 

1.  If  in  running  a  random    township  exterior, 
such  random  exceeds  or    falls  short  of  its   proper 
length  by  more  than  three  chains,  allowing  for  con- 
vergency,  or  falls  more  than  three  chains  to  the  right 
or  left  of  the  objective  point  (or  shows  a  proportion- 
ate error  for  lines  of  greater  or  less  length  than  six 
miles),  it  will  be  re-run,  and  if  found  correctly  run, 
so  much  of  the  remaining  boundaries  of  the  township 
will  be  retraced,   or  resurveyed,  as  may  be  found 
necessary  to  locate  cause  of  misclosure. 

2.  Every  meridional  section  line,   except   those 
which  terminate  upon  a  fractional  side  of  a  town- 
ship, will  be  80  chains  in  length,  without  allowance 
of  50  links   per  mile  for  difference  of  measure,  or 
any  other  allowance  beyond  a  small  reasonable  dis- 
crepancy according  to  the  nature  of  the  surface,  to 
be  determined  after  examination. 

3.  The  random   meridional  or    latitudinal   lines 
through  a  tier  or  range  of  fractional  sections  shall 
fall  within   50  links  of  the  objective  corners,  and  a 
greater  falling  will  indicate  negligence  or  error. 

4.  The  actual  lengths  of  meridional  section  lines 
through  a  fractional  north  or  south  tier  of  sections 
shall  be  within  150  links  of  their  theoretical  length. 
The  latter  will  be  determined  from  the  given  lengths 
of  meridional  boundaries  on  the  east  and  the  west 
range  lines. 

5.  Each    latitudinal    section    line,   except   in   a 
fractional  east  or  west  range  of  sections,  shall  be 
within  50  links  of  the  actual  distance  established  on 
the   governing   north    and    south  boundary  of    the 
township  for  the  width  of  the  same  range  of  sections. 

6.  The  north  boundary  and  the  south  boundary 
of  any  section,  except  in  a  fractional  range,  shall  be 
within  50  lengths  of  equal  length. 

7.  The  meanders  within  each  fractional  section 


256  A    MANUAL    OF    LAND    SURVEYING. 

or  between  any  two  successive  meander  corners,  or 
of  an  island  or  lake  in  the  interior  of  a  section, 
should  close  by  traverse  within  a  limit  to  be  deter- 
mined by  allowing  five  eighths  of  a  link  for  each 
chain  of  such  meander  line.  This  rule  does  not 
apply  to  irregular  boundaries  of  reservations  or  pri- 
vate claims,  except  as  far  as  the  same  are  natural 
water  boundaries.  The  total  naisclosure  of  meanders 
will  not  be  permitted  to  exceed  150  links,  except  in 
large  private  land  claims,  which  are  governed  by  a 
different  rule  and  limit. 

8.  In  closing  upon  accepted  surveys,  when  irreg- 
ularities beyond  the  allowable  limits  are  developed, 
either  in  the  length  or  direction  of  the  closing  lines, 
closing  corners  will  be  set  with  the   quarter-section 
corners  at   40  chains  from  the  last  interior  section 
corner. 

9.  And,    in    general,   when    conditions   are   met 
which  result  in  a  random  line  being  defective,  either 
in    length   or    direction,    such    procedure   will    be 
adopted  as  will  secure  the  greatest  number  of  new 
rectangular    legal    subdivisions,  without    disturbing 
the  condition  of  accepted  surveys. 

Field  Notes.— 1.  The  proper  blank  books  for  origi- 
nal field  notes  will  be  furnished  by  the  surveyor  general, 
and  in  such  books  the  deputy  surveyor  will  make  a 
faithful,  distinct,  and  minute  record  of  everything  done 
and  observed  by  himself  and  his  assistants,  pursuant  to 
instructions,  in  relation  to  running,  measuring,  and 
marking  lines,  establishing  corners,  etc.,  and  present, 
as  far  as  possible,  full  and  complete  topographical 
sketches  of  all  standard  and  exterior  lines,  drawn  to  the 
usual  scale  for  township  exteriors.  These  "original 
field  notes  "  are  not  necessarily  the  entries  made  in  the 
field,  in  the  deputy's  pocket  note  books  called  tablets; 
but  they  are  to  be  fully  and  correctly,  written  out  in 
ink,  from  such  tablets,  for  the  permanent  record  of  the 
work.  Tablets  should  be  so  fully  written  as  to  verify 
the  "original  field  notes  "  whenever  the  surveyor-gen- 
eral requires  them  for  inspection. 


ORIGINAL    SURVEYS.  257 

SURVEYING    BASE    LINES    AND     STANDARD      PARALLELS 
BY    OFFSETS    FROM    STRAIGHT    LINES. 

1.  The  corners  on  a  Base  Line  or  a  Standard 
Parallel  could  be  established  from  chords  of  the 
latitude  curve,  joining  successive  township  corners ; 
from  a  tangent  to  the  true  latitude  curve  at  a  point 
midway  between  the  township  corners  ;  or  from  an 
intermediate  straight  line  parallel  to  the  lines  above 
mentioned.  In  the  first  case,  all  the  offsets  would 
be  measured  south  ;  in  the  second  case,  all  offsets 
would  be  measured  north ;  and  in  both  cases,  the 
maximum  offset,  or  greatest  distance  of  the  latitude 
curve  from  the  reference  lines  would  be  one  fourth 
of  the  greatest  offset  from  a  tangent  six  miles  long, 
i.  e.,  the  offset  found  in  table  X,  opposite  the  proper 
latitude,  and  in  the  column  headed  ' '  3  miles  ; "  while 
LUC  initial  bearings  of  the  three  lines,  would  be 
equal  to  oach  other;  similar  relations  between  the 
bearings  at  corresponding  points,  would  obtain 
through  a  range. 

The  method  of  establishing  corners  on  a  true  lati- 
tude curve  by  offsets  from  a  line  situated  between 
and  parallel  to  the  chord  and  tangent,  which  was 
devised  to  meet  a  demand  for  short  offsets,  will  now 
be  described. 

Secant  Method. — i.  This  method  consists  of 
running  a  connected  series  of  straight  lines,  each  six 
miles  long,  on  such  courses  that  any  one  of  the  lines 
will  intersect  the  curve  of  the  parallel  of  latitude 
in  two  points  separated  by  an  interval  of  four  miles; 
and,  from  the  lines  thus  established,  measuring 
north  or  south,  as  the  case  may  be,  to  attain  other 
required  points  on  the  latitude  curve.  For  the  sake 
of  brevity,  said  straight  lines  will  be  called  secants 


UNIVERSITY, OF  CALIFORNIA. 

DEPA*TM««T  OF  CIVIL  *NCIN««*<NC 

•CRKC'.CY.  CALIFORNIA 


258 


A    MANUAL   OF    LAND    SURVEYING. 


TABLE  VIII.— Azimuths  of  the  Secant,  and  Offsets,  in  feet,  to  the  Parallel. 

Arguments;  latitude  in  left  hand  column  and  distance  from  starting  point  at  top  or  bottom  of  tb« 

table. 


Lati- 
tude. 

Azimuths  and  offsett'at— 

Deflec- 
tion Angle 
and  nat. 
tan.  to 
Rad.C6(t. 

0  miles. 

»  mile. 

1  mUe, 

1}  miles. 

2  mil**. 

2J  miles. 

3  miles. 

o 
30 

B# 

89*68',7 
0.87  K. 

89°69M) 
0.00 

89°69'.2 
0.67  S. 

89°  W.l> 
1.16  S. 

89°59'.7 
1.44  S. 

""*,?&,"»! 

S'00*.2 
0.69  tos. 

31 

89°  Iff.  i 
2.01  N. 

89°58'.6 
0.91  N. 

8*553 

89°  59'.2 
0.70  S. 

89°69'.5 
1.20  8. 

89°69'.7 
1.60  S. 

W(E-,?«W»! 

3'  07".4 
0.72  Ins. 

32 

89»Mf.4 
3.09  N. 

«9°58'.6 
0.94  N. 

«.*.. 

89°  59'  .2 
0.73  S. 

89°59'.6 
1.25  S. 

89°  6y.7 
1.66  S. 

W>(E.  orW.) 
1.67  Si 

3'  15".0 
O.Vilns. 

33 

r,fi? 

89°58'.6 
0.97  N. 

89°58'.8 
0.00 

89°  69M 
0.76  S. 

89°59'.4 
1.30  S. 

89°  59'.7 
1.62  S. 

90°  (E.  orW.) 
1.738. 

3'22".« 
0.78  Ins. 

34 

89°58'.2 
2.25  ?l. 

8fl°58'.5 

1.01  S. 

89°58'.8 
0.00 

89°  59M 
0.79  S. 

89°  69'.4 
1.36  S. 

89°  59'.7 
1.69  S. 

90°  (E.  or  W.) 
1*80  S* 

3'30».4 
0.81  Ins. 

'So 

89°  58'.2 
2*33  N. 

89<>58'.5 
1.05  N. 

-89t'58'.8 
0.00 

89°59M 
0.82  S. 

89°59'.4 
1.40  S. 

89°  W.l 
1.75vS. 

9(,(E.orW.) 

8'  38".4 
0.84  Ins. 

36 

89°68'.l 
2.42  N. 

89°58'.4 
1.09  N. 

WK 

89°  59'.0 
0.86  S. 

89°  69'.4 
1.46  S. 

89°  59'.7 
1.82  S. 

"Wi! 

3'  46".4 
0.87  las. 

3,7 

89°  68'.0 
3.51  X. 

89°  BO 
1.13  N. 

89°  68'.6 
0.00 

89°  68'.9 
0.88  S. 

89°  69'.3 
1.61  S. 

WW.l 
1.89  S. 

90°  (E.  orW.) 
2.01  s! 

3'  55".0 
0.90  Ins. 

38 

89°58'.0 
2.61  N. 

89°  58'.3 
1.17  N. 

89°68'.6 
0.00 

89°  68'.9 
0.91  S. 

89°  69'.3 
1.56  8. 

ea°  ^y.  7 

1.95  S. 

9V>(E2?08W8! 

4'  03".6 
0.03  Ins. 

30 

89°  57'.9 
2.70  N. 

89°  68'.2 
1.21  N. 

89°  58'.6 
0.00 

89°58'.9 
0.04  S. 

89°  591.3 
1.62  S: 

89°  69'.7 
2.02  8. 

^^Wn! 

4'  12".6 
0.97  Ins. 

40 

89°57'.8 
2.79  N. 

89°  58M 
1.25  N. 

89°  58'.5 
0.00 

89°  68'.  9 
0.98  8. 

89°  59'.3 
1.68  S. 

89°  59'.  7 
2.10  S. 

^(E.orWs) 

4'  2l".6 
1.00  Ins. 

41 

89°  57'.7 
2.89  N. 

89°68'.0 
1.30  N. 

89°  58'.4 

o.oo 

89°58'.8 
1.02  S. 

89°  59'.2 
1.74  8. 

89°  59'.  6 
2.17  8. 

""^WK 

4'  31".2 
1.04  in*. 

42 

89°  57'.  7 
3.00  N. 

89°  58'.0 
1.3d  N. 

"To* 

89°  58'.8 
1.06  S. 

89°  59'.2 
1.80  S. 

89°  59'.6 
2.26S. 

"•^Jtt 

4'  40":8 
1.1D8  In*. 

43 

•89°  57'.6 
3.11  N. 

89°  58'.0 
1.40  N. 

TcS 

8»°  58'.8 
1.08  S. 

89°5y.2 
1.86  S. 

89°59'.6 
2.33  S. 

90°  (E.  orW.) 
2.48  S. 

4'  50".8 
1.12ln». 

44 

89°  57'.5 
3.22  N. 

89°57'.9 
1.46  N. 

S^SS'.S 
0.00 

89°  58'.7 
1.12  S. 

89°69'.2* 
1.93  S. 

89°  59'.6 
2.41  S. 

90°  (E.  or  W.) 
2.67  S. 

6'  01"  0 
1.161ns. 

45 

89°  67'.4 
3.33  N. 

89°  57'.8 
1.60  X. 

89°  58',3 
0.00 

89°68'.7 
1.16  S. 

89°  59M 
2.00  S. 

89=  5y.5 
2.49  S. 

90°  (E.  or  W  ) 
2.66  S. 

5'11".8 
1.201ns. 

46 

89°  57'.3 
3.44  N. 

89°  57'.  7 
1.65  N. 

89°  68'.2 
0.00 

89°58'.6 
1.21  S. 

89°59M 
2.07  S. 

89°  59'.5 
2.59  S. 

90°  (E.  or  WO 
2.768. 

6'  22"  8 
1.24  Ins. 

47 

89°  57'  .2 
3.57  N. 

89°  57'.6 
1.61  N. 

89°58'.l 
0.00 

89°  58'.6 
1.25  S. 

89°  59M 
2.14  S. 

89°  69'  .5 
2.07<  S. 

90°  (E  orW.) 
2.868. 

6'  34".2 
1.28  Ins. 

48 

89°  57'.  1 
3.70  N. 

89°  57'.5 
1.66  N. 

89°  68'.  0 
0.00 

89°  58'.5 
1.30  S. 

89°59'.0 
2.22  S. 

89;>  69'  5 
2.78  S. 

""««?« 

5'  46".2 
1.331ns. 

49 

89°57'.0 
3.82  N. 

89°  67'.6 
1.72  N. 

89°58'0 
0.00 

89°68'.5 
1.34  S. 

89°  59'  0 
2.30  S. 

89°  59'.  5 
2.87  8. 

""Wi! 

5'  58".6 
1.38  Ins. 

50 

89°  55'.9 
3.96  N. 

89°  57'.4 
1.78  V. 

89°  57'.9 
0.00 

89°  58'.4 
1.39  S. 

89°  59'.0 
2.3S  S. 

89°  69'.5 
2.97  S. 

90°  (E.  or  W.) 

3.i7  s; 

6'11".4 
1.431ns. 

Lati- 
tude. 

6  miles. 

5i  miles. 

5  miles. 

41  miles. 

4  miles. 

31  miles. 

3  miles. 

Deflec- 
tion Angl* 
and  nat. 
tan,  to 
Bad.  66  ft. 

Azimuths  ^nd  offsets  at— 
'*;«>HJ           .  J  *Q  TM*M 

.t  J 


ORIGINAL    SURVEYS.  259 

2.  The  direction  of  the  first  secant  will  be  deter- 
mined at  its  initial  point  by  observations  on  Polaris 
at  elongation,  and  similar  observations  will  be  made 
at  intervals  not  exceeding  18  miles;  while  observa- 
tions by  the  method  given  on  page  96  et  seq  ,  or  on 
Polaris  at  elongation  (as  the  deputy  may  prefer),  will 
be  taken  every   night  when  practicable,  to   guard 
against    mistakes,    detect    errors,    and    check    the 
direction  of  the  line. 

The  principal  advantage  of  this  method,  over  that 
by  offsets  from  a  tangent,  results  directly  from  the 
proximity  of  the  secant  and  the  parallel  of  lati- 
tude, and  the  consequent  reduced  length  of  the 
maximum  offsets,  thereby  limiting  the  cutting,  which 
will  contain  both  secant  and  parallel,  to  a  single 
opening  less  than  four  feet  in  width  ;  avoiding  the 
necessity  for  clearing  out  roads  for,  and  instrument- 
ally  laying  off,  the  long  offsets  inseparable  from  the 
tangent  method  ;  and  permitting  the  noting  of  topo- 
graphical features  on  the  lines  actually  run,  a  con- 
venience not  always  attainable  by  the  tangent 
method. 

3.  In  any  given  case,  the  secant  lines  will  bear 
such  relation  to  the  latitude  curve,  that  points  on 
said  secants  at  one  and  five  miles  from  either  end  of 
any  secant,  will  be  coincident  with  two  points  on 
the  latitude  curve  four  miles  apart ;  between  which 
points  the  latitude  curve  will  lie  south  of  the  se- 
cants ;  while  the  curve  will  lie  north  of  the  secant 
lines  on   the  first  and  sixth  miles  ;  therefore  each 
secant  will  run  south  of  sees.  31  and  36,  in  every 
range,  and  through  all  other  sections  on  the  north 
side  of  the  base  line  or  standard  parallel,  as  the  case 
may  be. 

Each  secant,  the  azimuth  and  offsets  thereof,  and 
the  corresponding  part  of  the  parallel,  will  be  sym- 
metrically divided  by  the  middle  meridian  of  each 
range,  i.  e.,  the  bearings  and  offsets  at  equal  dis- 
tances on  opposite  sides  of  the  central  meridian  will 


260  A    MANUAL    OF    LAND    SURVEYING. 

be  equal;  the  bearings,  which  continually  change, 
will  always  be  north  of  east  (or  west),  on  the 
first  three  miles,  and  south  of  east  (or  west) ,  on  the 
last  three  miles  of  each  secant.  The  changes  of 
bearing  should  not  be  understood  to  imply  a  change 
of  direction  of  any  secant  with  respect  to  its  initial 
direction  ;  the  change  is  due  to  the  varying  inclina- 
tion of  the  meridians  to  the  straight  secant,  i.  e. , 
the  effect  of  convergency  of  meridians. 

4.  Employing  the  data  provided  by  Table  II,  the 
practical  application  of  the  method  herein  outlined 
will  be  conducted  in  the  field  as  follows  : 

Set  up  the  carefully  adjusted  transit  south  of  the 
township  corner  at  which  the  survey  will  begin,  and 
at  a  distance  therefrom  to  be  interpolated  for  the 
given  latitude,  from  the  column  headed  "0  miles." 
in  Table  VIII.  By  observations  on  Polaris  at 
elongation,  determine  and  mark  a  true  meridian. 

Lay  off  the  azimuth,  found  in  the  table  under  "  0 
miles,  "  toward  the  east  (or  west),  as  the  case  may 
be,  and  re-measure  the  angle  a  sufficient  number  of 
times  to  secure  an  accurate  result. 

Produce  the  direction   of  the  secant  thus  deter- 
mined, a  distance  of  six  miles  in  a  straight  line, 
taking  double  back  and  fore  sights  at  each  setting  of 
the  instrument.     At  each  half-mile  and  mile  point, 
establish  on  the  standard  parallel  the  proper  quarter 
section    and    section    corners,    by   offsets  of  correct 
length,  north  or  south,  as  indicated  in  the  table  by 
the  initial  letters  N.  or  S. 

The  offsets  being  very  short,  their  direction  (per- 
pendicular to  the  secant,  without  sensible  error), 
may  be  determined  by  the  eye  ;  the  length  of  offsets 
should  be  carefully  measured. 

At  6  miles  on  the  secant,  turn  off  to  the  north  the 
proper  deflection  angle,  given  in  the  right  hand 
column  of  the  table,  thereby  defining  the  direction 
of  a  new  secant,  from  which  points  will  be  estab- 


OBIGIXAL    SURVEYS.  261 

lished  on  the  parallel,  as  directed  in  preceding  para- 
graph. 

5.  Applications  of  Table  VIII. — The  true  bear- 
ing of  the  secant  at  each  mile  and  half-mile  point 
will  be  expressed  by  the  tabular  azimuth  preceded 
by  the  initial  meridional  letter  X.,  when  the  distance 
argument  is  found  at  the  top  of  the  table  ;  but  when 
said  argument  is  found  at  the  bottom  of  the  table, 
the  meridional  letter  S.  will  be  placed  before  the 
azimuth;  while  the  departure  letter,  E.  or  W.,  will 
be  made  to  agree  with  the  direction  of  the  survey, 
east  or  west,  as  the  case  may  require.  The  bearings 
will  be  taken  from  the  table,  to  the  nearest  whole 
minute  only,  and  entered  at  the  beginning  of  each 
mile  recorded  in  the  field  notes.  The  direction  of 
the  offsets  or  distances  from  the  secant  north  or 
south  to  the  base  line  or  standard  parallel,  as  the 
case  may  be,  are  indicated  by  the  initial  letters,  X. 
or  S.  following  the  offsets. 

Example  1. — Standard  parallel  run  west,  lat.  48° 
N.  ;  dist.  from  initial  point  of  secant,  2  miles  ;  the 
bearing  is  X.  89°  59'  W.,  the  offset,  2.22  ft.,  S.  ;  at 
5^  miles  the  bearing  is  S.89°  57'  W.,-the  offset  1.66 
ft.  X.  In  all  latitudes  the  bearing  of  the  secant  at 
3  miles  will  be  east  or  west,  agreeing  with  the 
direction  of  the  parallel. 

The  offsets  may  be  interpolated  for  minutes  of 
latitude,  by  simple  proportion,  as  follows:  Multiply 
thf  difference  between  the  offsets  corresponding  to  the 
whole  degrees  of  latitude,  immediately  preceding  and 
following  the  given  latitude,  by  the  minutes,  ex- 
pressed in  decimals  of  a  degree,  and  add  the  prod- 
uct to  the  offset  corresponding  to  the  lesser  lati- 
tude ;  the  sum  will  be  the  offset  required. 

Example  2  , — Lat.  45°  34'  ,5  ;  dist.,  0  miles  or  6 
miles  ;  the  diff.  between  offsets  in  latitudes  45°  and 
46°  is  0.11  ft.  ;  34 '.5=0. 575  ;  0.11X0.575— 0.06  ft.  ; 
and,  3.33+0.06=3.39  ft.  the  offset  required.  A 

; 


262  A    MANUAL    OF    LAND    SURVEYING. 

similar  method  of  interpolation  may  be  applied  to 
the  data  in  the  right-hand  column. 

Example,  S. — Latitude  45°  34 '.5;  diff.  of  angles 
isO'  11";  11 X 0.575=6". 3  ;•  and  5'  11". 8 +  6". 3= 
5'  18",  nearly;  also  0'. 04X0.575=0  02  ins.';  and 
1.20+0.02=1.22  ins. 

6.  The  deputy  should    clearly  understand  from 
the  foregoing  rules  and  directions  that  the  correct 
establishment  of  a^  standard  parallel  on  a  true  latitude 
curve,  by  offsets  from  secant  lines,  will  depend  in 
the  order  of  sequence  upon  careful  attention  to  the 
following  points  : 

1.  Accurate  observations  on  Polaris  at  elonga- 
tion, to  determine  a  true  meridian. 

2.'  Close  measurement  of  the  azimuth  angle,  to 
define  the  initial  direction  of  the  secant. 

3.  Careful  prolongation  of  the  secant  in  a 
straight  line. 

.  4.     Correct  measurement  of  the  deflection  angle 

7.  With  ordinary  field  instruments,  usually  read- 
ing to   single  minutes  only,  fractional  parts  of  the 
"least    count"    are  generally  estimated  by  the  eye 
Greater  accuracy  may  be  obtained  by  making  use  of 
a  linear  measure  to  lay  off  deflection  angles.     Table 
VIII  supplies  the  requisite  data  ;   "the  natural  tan- 
gent of  the  angle  of  deflection  to  a  radius  of  one 
chain,"  inserted  in  the  right-hand  column,  may  be 
employed  as  follows  : 

Having  taken  a  back  sight  at  the  6-mile  point  on 
the  secant,  at  exactly  one  chain  in  advance  of  the 
center  of  the  instrument,  place  upon  the  ground  in 
a  horizontal  position,  and  precisely  at  right  angles 
to  the  line,  a  rule  or  scale  divided  into  decimal  parts 
of  an  inch,  move  the  scale  north  or  south  until  one 
of  its  principal  lines  appears  coincident  with  the  ver- 
tical wire  ;  then  with  the  tangent  screw  of  the  ver- 
nier plate,  carry  the  wire  over  the  scale  toward  the 
north,  the  required  distance,  i.  e. ,  the  length  of  tan- 
gent in  the  right-hand  column.  The  readings  of  the 


ORIGINAL    SURVEYS.  263 

vernier   will    check   the    measurement    and    guard 
against  mistakes. 

A  piece  of  white  paper  with  two  fine  parallel  lines 
drawn  across  it,  exactly  the  proper  distance  apart, 
pasted  on  a  thin  slip  of  wood  (such  as  a  piece  of 
cigar  box  3  inches  long  by  1  inch  wide),  will  make 
an  accurate  and  very  convenient  and  portable  sub- 
stitute for  a  rule  or  scale.  Several  copies  may  be 
prepared  in  advance  to  replace  the  original  in  case 
of  loss. 

8.  To  mark  the  direction  of  the  new  secant  thus 
determined,  set  a  flag  on  line,  and  as  far  in  advance 
of  the   instrument   as   practicable.       The   direction 
will  be  verified  by  another  similar  observation,  to  be 
made  after  revolving  the  azimuth  circle  180°. 

Theoretically,  it  is  immaterial  whether  the  scale 
tye  placed  above  or  below  the  level  of  the  tele- 
scope provided  the  horizontal  distance  from  the 
center  of  the  instrument  is  accurately  one  chain 
(66  ft.);  practically,  the  most  satisfactory  result  will 
be  had  on  level  ground,  suitable  for  correct  meas- 
urement of  the  distance. 

9.  The    secant    method    adapted  to  transit  in- 
struments exclusively,  is  recommended  for  its  sim- 
plicity and  accuracy,  and  the  facility  with  which 
the  line  may  be  extended  over  rough  mountainous 
land  or  through  dense  undergrowth  ;  in  deep  valleys 
or  canyons  where  the  sun  cannot  be  observed    in 
favorable  positions  ;  or  anywhere  during  the  contin- 
uance of  adverse  weather  conditions  and  under  cir- 
cumstances when  the  use  of  solar  apparatus  would 
be,  if  not  impossible,  at  least  inconvenient  and  un  - 
reliable. 

10.  The  true  bearing  of  a  line  joining  any  two 
points  on  a  standard  parallel  will  be  obtained  from 
Table  IX,  by  taking  it  from  the  column  headed  with 
one-half  of  the  distance  between  said  points.     Ex- 


204  A    MANUAL    OF    LAND    SURVEYING. 

ample.  Required  the  bearing  from  corner  of  sees. 
32  and  33,  R.  22  E.,  to  corner  of  sees.  32  and  33 
E.,  R.  21  E.  The  latitude  is  45°  '34'  .5,  the  dis- 
tance 6  miles.  Consequently,  the  azimuth  from  the 
column  marked  "3  miles"  for  the  given  latitude,  is 
N.  81J°  57'  20"  .9  W.,  the  required  true  bearing. 

Tangent  Method. —  This  method  consists  in  lay- 
ing off  from  a  true  meridian,  established  by  obser- 
vations on  Polaris  at  elongation,  an  angle  of  90°  pro- 
ducing the  directions  thus  determined,  a  distance  of 
6  miles  in  a  straight  line,  and  measuring  north 
therefrom,  at  half-mile  intervals,  distances  of  correct 
length,  taken  from  Table  X  (interpolated  if  neces- 
sary), for  the  given  latitude,  to  attain  other  points 
on  the  latitude  curve  passing  through  the  tangential 
or  initial  point. 

The  azimuth  or  bearing  of  the  tangent  at  succes- 
sive mile  points  will  be  taken  from  Table  IX  to  the 
nearest  whole  minute  only,  and  will  be  inserted  in 
the  field  notes,  no  interpolation  being  required,  ex- 
cept when  test  sights  are  taken.  The  true  bearing 
between  two  points  on  a  standard  parallel  will  be 
derived  from  Table  IX  by  taking  it  in  the  column 
headed  with  one-half  the'  distance  between  said 
points.  The  offsets  at  intervals  of  one  mile  are  in- 
serted in  Table  X  ;  to  obtain  the  length  of  offsets  at 
the  half-mile  points,  take  one  fourth  the  offset  cor- 
responding to  twice  the  distance  of  the  half-mile, 
point  from  the  tangential  point. 

This  method  is  suitable  for  running  standard  par- 
allels and  latitudinal  township  lines  in  a  level  open 
country,  where  no  intersections  with  topographical 
features  will  be  required  ;  bu|  in  all  cases  the  secant 
method  will  be  found  most  convenient. 


ORIGINAL    SURVEYS. 


265 


TABLE  IX.— Azimutlis  oftJie  Tangent  to  the  Parotid. 

fThe  azimuth  IB  the  smaller  angle  the  tangent  make's  with  the  true  meridian  and  always  measured 
from  the  north  and  towards  the  tangential  points.] 


Lati- 
tude. 

1  mile. 

2  miles. 

3  miles. 

4  miles. 

6  nlles. 

6  miles. 

80 
II 
88 

11 

89  59  30.0 
89  59  28.8 
89  69  27.5 

89  59  26.2 
89  59  24.9 
89  59  23.6 

89  58  69.9 
89  58  57.5 
89  58  55.0 

89  58  62.5 
89  58  49.9 
89  68  47.2 

89  53  29.9 
89  5S  2C.3 
89  68  22.5 

89  68  18\7~ 
89  5tf  14.8 
89  56  10.8 

89  67  69.9 
89  57  66.0 
89  57  60.0 

89  67  44.9 
89  57  39.7 
89  57  34.4 

89  67  29.9 
89  57  23.8 
89  67  17.6 

89  67  U.2 
89  57  04.6. 
89  66  68.0. 

89  56  59.8 
89  66  52.6 
89  66  4&.0 

89  66  37.4 
89  56  29.S 
89  66  21.6 

1 

89  59  22.2 
89  09  20.8 
89  59  19.4 

89  68  44.4 
89  58  41.6 
89  68  38.8 

89  58  06.8 
89  68  02.5 
89  57  58.2 

89  57  28.9 
89  57  23.3 
89  57  17.5 

89  56  61.1 
S9  56  44.1 
89  66  36.9 

89  66  13.4 
89  56  05.Q 
89  55  56.3 

80 
40 
41 

89  59  17.9 
89  59  16.4 
89  59  14.8 

89  58  35.8 
89  58  32:8 
89  58  29.6 

89  57  63.7 
89  57  49.2 
89  57  44.4 

89  67  11.6 
89  6T  C5.5 
89  56  59.3 

89  56  29.6 
89  56  21.9 
89  56  14.1 

89  55  47.  & 
89  55  38.8 
89  55  28.  » 

48 

48 
M 

89  59  13.2 
89  59  11.5 
89  59  09.8 

«9  58  26.4 
89  58  23.1 
89  58  19.6 

89  57  39.6 
89  57  34.6 
89  67  29.5 

89  56  52.8 
89  56  46.2 
89  56  39.3 

89  56  06.0 
89  55  67.7 
89  55  49.1 

89  55  19.2 
89  55  09.2 
89  54  68.9 

45 

46 

47 

89  59  08.0 
89  59  06.2 
89  59  04.3 

89  58  16.1 
89  58  12.4 
89  58  08.6 

89  67  24.1 
89  57  18.6 
89  57  1£.9 

89  56  32.1 
89  56  24.8 
89  56  17.1 

89  55  40.2 
89  55  31.0 
89  55  21.4 

89  64  48.2 
89  54  37.2 
89  54  25.7 

M 
49 
50 

89  59  02.3 
89  69  00.2 
8?  68  58.1. 

89  58  04.6 
89  58  00.5 
89  57  56.2 

89  57  06.9 
89  57  00.7 
89  56  54.3 

89  56  09.2 
89  K  00.9 
89  56  W.« 

89  55  11.5 
89  55  01.2 
89  54  50.5 

89  54  13.8 
89  M  01.4 
89  63  48.5 

L»tl- 
tmde. 

7  miles. 

8  miles. 

9  miles. 

Id  miles. 

11  miles. 

12  miles. 

H 

89  56  29.8 
89  66  21.3 
89  56  12.5 

89  M  59.8 
89  u5  60.0 
89  65  40.0 

89  55  29.3 
89  55  18.8 
89  66  07.6 

89  54  59.7 
89  54  47.6 
89  54  35.1 

89  &t  29.7 
89  54  16.3 
89  54  02.6 

89  53  59.7 
89  53  45.1 
89  63  30.1 

& 

89  55  03.6 
89  55  64.5 
89  66  45.2 

89  55  29.9 
89  55  19.4 
89  66  08.8 

89  64  56.1 
89  54  44.4 
89  64  32.3 

89  M  2-2.3 
89  54  09.3 
89  63  65.9 

89  58  48.5 
89  53  34.2 
89-  63  19.5 

89  G3  14.8 
89  52  63.1 
89  62  43.1 

80 

11 

89  65  35.6 
89  66  25.8 
89  69  15.7 

89  54  67.8 
89  54  4&6 
89  64  36.1 

89  64  20.0 
89  64  07.4 
89  63  64.6 

89  63  42.3 
89  63  2&2 
89  63  13.9 

89  53-  04.6 
89  52  49.1 
89  62  33.2 

89  52  26.7 
89  52'09.t 
89  51  52.6 

% 
41 

iv-.l 
42 
43 
44 

89  66  06.4 
89  64  64.7 
89  64  43.7 

89  54  32.4 
89  64  20.8 
89  54  08.7 

89  54  23.3 
89  64  11.1 
89  63  68.5 

89  53  45.6 
89  63  32.3 
89  63  18.5 

89  63  41.2 
89  63  27.5 
89  63  13.4 

89  62  68.8 
89  52  43.8 
89  62  28.4 

89  62  59.1 
89  62  43.8 
89  62  28.2 

89  52  12.0 
89  61  'b6.4 
89  61  38.2 

89  62  17.0 
89  62  00.2 
89  61  43.0 

89  51  26.2 
89  61  06.9 
89  50  48.0 

89  51  34.  1 
89  61  16.  < 
88  50-S7.  J 

89  50  38.  4 
89  60"'18.6 
89  49  57.8 

46 
46 
47 

89  53  66.3 
89  53  43.4 
89  53  30.0 

89  53  04.3 
89  52  49.5 
89  52  34.3 

89  52  12.3 
89  61  65.7 
89  51  38.6 

89  51  20.4 
89  51  01.9 

89  50  42.9 

89  60  28.4 
89  50  08.1 
89  49  47.2 

89  49  36.4 
89  49  14.  S 

89  48  51.4 

43 

IS 

89  63  16.1 
89  63  01.7 
89  62  46.6 

89  52  18.4 
89  52  01.9 
89  61  44.7 

89  61  20.7 
89  51  02.1 
89  50  42.8 

89  60-23.0 
89  60  02.4 
89  49  40  9 

89  49  25.3 
89  49  02.6 
89  48  39.0 

89  48  27.C 
89  48  02.8 
89  47  S7.1 

283  A   MANUAL    OF    LAND    SURVEYING. 

TABLE  X.— Offsets,  in  Chains,  from  Tangent  to  Parallel. 


Utl- 
tudft. 

1  mile. 

2  mile*. 

3  miles. 

4  miles. 

6  miles. 

«  miles. 

sr 

Si 

Chain*. 
0.006 
0.006 
0.006 

Chains.     • 
0.023 
0.024 
0.025 

Chains. 
0.053 
0.066 
0.057 

CAain*. 
0.09 
0.10 
0.10 

Chains. 
0.14 
0.15 
0.16 

Chains. 
0.21 
0.22 
0.23 

H 

0.007 
0.007 
0.007 

0.026 
0.027 
0.028 

0.059 

tss 

o.lo 

0.11 
0.11 

0.16 
0.17 
0.18 

0.24 
0.26 
0.26 

•1 

0.007 
0.008 
0.008 

0.029 
0.031 
0.032 

0.066 
0.068 
0.071 

0.12 
0.12 
0.13 

0.18 
0.19 
0.20 

0.26 

S:S 

8 

41 

0.008 
X 

0.033 

£$ 

0.074 
0.076 
0.079 

0.13 
0.13 
0.14 

0.20 
0.21 
0.22 

0.29 

o°:H 

42 
S 

•&8S 

0.010 

0.036 
0.038 
0.039 

0.082 

£S 

0.14 

0.15 
0.16 

0.23 
0.24 
0.24 

i2 

0.36 

46 

t? 

0.010 
0.010 
0.011 

0.040 
0.042 
07044 

0.091 
0.094 
0.097 

0.16 
0.17 
0.17 

0.25 
0.26 
0.27 

0.36 
0.37 
0.39 

48 
49 
60 

0.011 
0.012 
0.012 

0.046 
0.046 
0.048 

0.101 
0.104 
0.108 

0.18 
0.19 
0.19 

0.28 
0.29 
0.30 

0.40 
•      0.42 
0.43 

Latl. 
tude. 

7  miles. 

8  mile*. 

9  miles. 

10  miles. 

11  miles. 

12  mll«t. 

80 
81 
32 

Chains. 
0.29 
0.30 
0.31 

Chains. 
0.37 
0.39 
0.40 

Chains. 
0.47 
0.49 
0.61 

Chains. 
0.56 
0.60 
0.63 

Chains. 
0.71 
0.74 
0.76 

Chnin*. 
0.84 
0.88 
0.91 

g 

0.32 
0.33 
0.36 

0.42 
0.43 
0.46 

ts 

0.67 

0.66 
0.68 
0.70 

0.79 
0.82 
0.86 

0.95 

ts 

8 

88 

0.36 
0.37 
0.38 

0.17 
0.48 
0.60 

0.69 

0.61 
0.64 

0.73 
0.76 
0.78 

0.89 
0.91 
0.96 

1.06 
1.10 
1.14 

89 

» 

8:J? 

0.43 

0.62 
0.64 
0.66 

0.66 
0.68 
070 

0.81 
0.84 
0.87 

0.99 
1.02 
1.06 

1.18 
1.22 
1.26 

8 

0.44 

0.46 
0.48 

0.68 

S:§ 

0.73 
0.76 
0.79 

0.90 

si 

1.09 
1.14 
1.18 

1.31 
1.36 
1.40 

ts 

47 

0.49 
0.51 
0.63 

0.64 

£8 

0.81 
0.84 
0.87 

1.00 

!:S 

1.22 
1.26 
1.31 

1.46 

1.60 
1.66 

48 

X 

0.66 

IS 

|3 

0.77 

0.91 

53 

1.12 
1.16 
1.20 

1.36 
11 

1.61 
1.67 
1.73 

SUBDIVISION  OF  SECTIONS.  267 


CHAPTEE  IX. 

SUBDIVISION  OF   SECTIONS. 

1.  Subdivisions  of  sections  are  original  surveys 
to  be  made  in  the  following  manner: 

1.  Section  and  quarter-section  corners  set  by  the  gov- 
ernment surveyors,  and  the  boundaries  actually  run  by 
them,  as  well  as  the  length  of  all  lines  as  returned  in 
their  field  notes,  are  to  be  taken  as  correct.    (See  Sec.  2396 
R.  S.,  First  and  Second.    P.200,  Sec.  100.) 

2.  The  corners  of  half  and  quarter  sections  which  were 
not  marked  on  the  government  surveys,  must  be  placed 
•as  nearly  as  possible  equidistant  from  those  two  comers 
which  stand  on  the  same  line.    (Sec.  2396,  First.    P.200, 
Sec.  100.) 

This  applies  to  the  quarter-posts  on  the  north  and  west 
lines  of  the  township  which  were  surveyed  previous  to 
1846;  also  to  those  townships  which,  under  the  act  of  1796, 
were  surveyed  into  blocks  of  two  miles  square  (P.200. 
Sec.  99,  Third),  and  to  those  surveyed  under  the  act  of 
1800,*  where  no  quarter-section  corners  were  planted  on 
the  lines  running  from  south  to  north. 

*No.  21.— An  Act  to  amend  the  act  entitled  "An  act  providing  for 
the  sale  of  the  lands  of  the  United  States,  in  the  territory  northwest 
of  the  Ohio,  and  above  the  mouth  of  the  Kentucky  River." 

SEC.  3.  And  be  it  further  enacted.  That  the  surveyor-general  shall 
cause  the  townships  ^west  of  the  Muskingum,  which  by  the  above- 
mentioned  act  are  directed  to  be  sold  in  quarter  townships,  to  be  sub- 
divided into  half  sections  of  three  hundred  and  twenty  acres  each,  as 
nearly  as  may  be,  by  running  parallel  lines  through  the  same  from  east 
to  west,  and  from  south  to  north,  at  the  distance  of  one  mile  from  each 


'268  A  MANUAL  OF  LAND  SURVEYING. 

3.  The  boundary  lines  of  sections,  (see  Page  198,  Sec.  99, 
Third),  and  of  half  and  quarter  sections,  which  were  not 
actually  run  and  marked,  are  to  be  ascertained  by  run- 
ning straight  lines  from  the  established  corners  to  the 
opposite  corresponding  corners.  Where  no  such  opposite 
corners  have  been  or  can  be  fixed,  the  line  should  be  run 
from  the  established  corner  due  north  and  south  or  east 
and  west,  as  the  case  may  be,  to  the  water-course  or 
other  external  boundary.  (P.200,  Sec.  100,  Second.)  These 
due  lines  are  to  be  found  by  trial  of  the  boundary  lines 
of  the  section,  as  actually  run  by  the  government  sur- 
veyor, and  the  subdivision  line,  run  on  a  course  interme- 
diate between  the  courses  of  the  section  lines  which  lie 
parallel  with  it. 

The  following  figure  illustrates  the  manner  of  sub- 
dividing sections.  It  shows  sections  5,  6,  7,  and  8,  repre- 

other,  and  marking  corners,  at  the  distance  of  each  half  mile  on  the 
lines  running  from  east  to  west,  and  at  the  distance  of  each  mile  on 
those  running  from  south  to  north,  and  making  the  marks,  notes,  and 
descriptions  prescribed  to  surveyors  by  the  above-mentioned  act :  And 
the  interior  lines  of  townships  intersected  by  the  Muskingum,  and  of 
all  the  townships  lying  east  of  that  river,  which  have  not  been  hereto- 
fore actually  subdivided  into  sections,  shall  also  be  run  and  marked  in 
the  manner  prescribed  by  the  said  act  for  running  and  marking  the 
interior  lines  of  townships  directed  to  be  sold  in  sections  of  six  hun- 
dred and  forty  acres  each.  And  in  all  cases  where  the  exterior  lines 
of  the  townships,  thus  to  be  subdivided  into  sections  or  half-sections,- 
shall  exceed  or  shall  not  extend  six  miles,  the  excess  or  deficiency 
shall  be  specially  noted,  arid  added  to  or  deducted  from  the  western 
and  northern  ranges  of  sections  or  half-sections  in  such  township,  ac- 
cording as  the  error  may  be  in  running  the  lines  from  east  to  west,  01 
from  south  to  north;  the  sections  and  half-sections  bounded  on  the 
northern  and  western  lines  of  such  townships  shall  be  sold  as  contain- 
ing omy  the  quantity  expressed  in  the  returns  and  plats,  respectively, 
and  all  others  as  containing  the  complete  legal  quantity.  And  the 
President  of  the  United  States  shall  fix  the  compensation  of  the  dep- 
uty surveyors,  chain-carriers,  and  axemen:  Provided,  The  whole  ex- 
pense of  surveying  and  marking  the  lines  shall  not  exceed  three 
dollars  for  every  mile  that  shall  be  actually  run,  surveyed,  and 
marked. 


SUBDIVISION"  OF    SECTIONS. 


269 


sentiiig  the  four  different  cases  which  occur  in  a  township 
surveyed  previous  to  1846.  In  the  later  surveys,  the  de- 
TVj^n ._s_h jj;  J  i  n  B  tails  would  dif- 

fer a  little,  ow- 
ing to  the  fact 
that  the  section 
and  quarter- 
section  corners 
on  the  town- 
ship and  range 
lines  are  com- 
mon to  the 
townships  o  n 
each  side  of  and 
ad  joining  those 
lines.  The  prin- 
ciple of  subdi- 
vision is,  how- 


3  '  * 

t~'^'~l 

*_.  j_J^S-2.  1 

_  .  JLet; 

3992, 

;  F  * 

o  «  V 

1  1  " 

•f+0.  r>0  \20flO  [J./.33 

*  UJ7  * 

5C. 

A         J 

!  \  I 

11  26  £».••  ^o  oo 

39.R9, 

c 
39.  as. 

FIG.  69. 

CASE  1.— Section  8.  All  the  quarter  posts  are  at  equi- 
distant points  from  the  section  corners  which  are  on  the 
same  line. 

CASE  2. — Section  5.  Quarter  posts  on  the  north  and 
the  south  are  at  equidistant  points.  Those  on  the  east 
and  the  west  are  40  chains  from  the  south  line  of  the  sec- 
tion. The  fraction  is  on  the  north  half  of  the  section. 

CASE  3. — Section  7.  Quarter  posts  on  the  north  and 
the  south  are  placed  at  40  chains  from  the  east  line  of  the 
section.  Those  on  the  east  and  the  west  are  at  equidistant 
points.  The  west  half  of  the  section  is  fractional. 

CASE  4. — Section  6.  The  quarter  posts  on  the  north 
and  the  south  are  placed  at  40  chains  from  the  east  line 
of  the  section.  Those  on  the  east  and  the  west  are  40 
chains  from  the  south  line  of  the  section.  Fractional 
both  on  the  north  and  west. 

NOTE.— In  1856,  Thomas  A.  Hendricks,  then  Commissioner  of  the 
General  Land  Office,  gave  the  following  rule  for  locating  the  center  of 
a  section:  "Run  a  true  line  from  the  quarter-section  corner  on  the 
east  boundary,  to  that  in  the  west  boundary,  and  at  the  equidistance 
between  them  establish  the  corner  for  the  center  of  the  section." 


270  A  MANUAL   OF  LAND   SURVEYING. 

This  was  in  harmony  with  an  opinion  previously  given  by  the  Sur- 
veyor General  of  Missouri  and  Illinois,  and  was  very  generally  fol- 
lowed by  the  surveyors  in  those  States.  This  rule  has  not  been  sus- 
tained by  the  courts,  nor  by  any  other  ruling  of  the  Land  Office,  so  far 
as  we  can  learn.  It  was  expressly  overruled  by  the  Secretary  of  the 
Interior  in  1868. 

Quarter- sections  are  to  be  subdivided  into  half -quar- 
ters by  lines  running  north  and  south. 

The  corners  which  were  not  marked  are  to  be  placed  as 
nearly  as  possible  equidistant  between  the  two  corners 
of  the  quarter-section  which  stand  on  the  same  line. 
Then  run  straight  lines  from  the  established  corners  to 
the  opposite  corresponding  corners,  (Page  202,  Sec.  101.) 

Half-quarter  sections  are  to  be  subdivided  into  quar- 
ter-quarters in  a  similar  manner,  by  east  and  west  lines, 
(P.  202,  Sec.  101.) 

It  may  be  well  to  remark  here,  that  the  instructions  from  the  Gen 
eral  Land  Office  have  not  been  uniform  in  regard  to  the  proper  manner 
of  subdividing  quarter-sections,  and,  as  might  be  expected,  the  prac- 
tice is  not  uniform  among  good  surveyors  Commissioners  Wilson 
and  Edmunds  held  that  half-quarter  and  quarter-quarter  lines  should 
be  "  straight  lines  running  through  the  section  "  to  points  on  the  sec- 
tion line.  (See  Hawes's  Manual,  p.  142,  and  Dunn's  Land  Laws,  p,  19.) 

The  foregoing  rules  are  those  of  the  statute,  and  are  endorsed  by 
Commissioners  Drummond,  Williamson,  and  McFarland. 

Commissioner  Drummond's  instructions  are  as  follows: 

"  In  the  subdivision  of  quarter-sections,  the  quarter-quarter  posts 
are  to  be  placed  at  points  equidistant,  and  on  straight  lines  between 
the  section  and  quarter-section  corners,  and  between  the  quarter-cor- 
ners and  the  common  center  of  the  section,"  etc.  The  difference  in  the 
two  methods  occurs  when,  as  very  often  happens,  the  quarter-posts 
are  not  in  line  between  the  section  corners. 

2,  Fractional  sections  are  to  be  subdivided  ac- 
cording to  the  Fifth  paragraph  of  Sec.  2395  of  the  Kevised 
Statutes,  under  such  rules  and  regulations  as  may  be  pre- 
scribed by  the  Secretary  of  the  Interior.  (Sec.  99,  Ex. 
Land  Laws,  and  TJ.  S.  Instructions,  1881,  p.  39.) 

Under  these  regulations.*  the  fractional  quarter-sections 
lying  next  to  the  north  line  of  the  township  are  divided 

*  NOTE.—"  Circular  to  Surveyors-General,  Nov.  9,  law.— SIR:  By  the 
first  section  of  the  act  of  April  24, 1820,  all  the  public  lands  of  the  Uni- 
ted States  shall  be  offered  at  public  sale  in  half-quarter  sections;  and 


SUBDIVISION  OF  SECTIONS.  27 1 

into  half -quarters  by  lines  running  east  and  west,  parallel 
with  and  twenty  chains  distant  from  the  quarter-section 
line.  (See  Keasling  v.  Truitt,  30  Ind.  506.) 

The  quarter- sections  lying  next  to  the  west  line  of  the 
township  are  divided  into  half-quarters  by  lines  running 
north  and  south,  parallel  with  and  twenty  chains  distant 
from  the  quarter-section  line. 

3 .  Section  6  adjoins  both  the  north  and  the  west 
lines  of  the  township,  and  is  subject  to  both  rules.  The 
north  half  is  divided  into  half -quarters  by  an  east  and 
west  line,  and  the  south  half  by  north  and  south  lines. 

The  quarter-post  on  the  north  side  of  section  six  should 
be  placed  on  the  township  line  at  a  point  40  chains  of 
original  measure  west  from  the  northeast  corner  of  the 
section. 

The  quarter-post  on  the  west  line  of  section  six  should 
be  placed  at  a  point  on  the  range  line  40  chains  of  orig- 
inal measure  north  from  the  southwest  corner  of  the 
section.  By  anginal  measure  is  meant  such  measure  as 
was  actually  laid  down  on  the  ground  by  the  deputy  sur- 
veyors who  made  the  original  survey. 

fractional  sections  containing  one  hundred  and  sixty  acres  and  up- 
ward shall,  as  nearly  as  practicable,  be  divided  into  half-quarter  sec- 
tions, under  such  rules  and  regulations  as  m&y  be  prescribed  by  the 
Secretary  of  the  Treasury;  but  fractional  sections  containing  less  than 
one  hundred  and  sixty  acres  shall  not  be  divided,  etc.  By  the  act  of 
May  10, 1800,  section  3,  the  excess  or  deficiency  of  regular  sections  or 
quarter-sections  in  any  township  is  to  be  thrown  on  the  north  and 
west  sides  of  the  township,  making  fractional  sections  more  or  less 
than  one  hundred  and  sixty  acres.  In  subdividing  such  fractional 
sections  to  form  a  half-quarter  section,  viz.,  80  acres,  the  Secretary  of 
tha  Treasury  directs  that  the  subdividing  line  for  such  fractions  as  lie 
on  the  north  side  of  a  township  sha.ll  be  an  east  and  west  line,  forming 
the  half-quarter  section  on  the  south  side  of  the  fraction;  and  for  such 
fractions  as  lie  on  the  west  side,  the  subdividing  line  shall  be  a  merid- 
ian, forming  the  half-quarter  section  on  the  east  side  of  the  fraction. 
This  mode  of  subdivision  will  preserve  the  compactness  of  the  tracts 
with  the  general  divisions,  and  will  not  interfere  with  the  rule  adopted 
relative  to  fractions  formed  by  a  stream,  a  river,  etc." 


272 


A  MANUAL  OF   LAND   SURVEYING. 


Iii  further  subdividing  the  northwest  quarter  of  Section 
6  into  quarter-quarters,  it  is  done  by  a  line  parallel  with 
and  20  chains  west  of  the  north  and  south  quarter -sectipn 
line. 

The  foregoing  is  the  general  plan  adopted  for  the  sub- 
division of  sections  of  the  United  States  Survey.  There 
have,  however,  been  many  exceptions  in  the  earlier  official 
plats,  in  accordance  with  which  the  land  was  sold.  To 
meet  all  such  cases  the  rule  has  been  adopted  to  subdivide 
in  such  a  way  as  to  suit  the  calculation  of  the  areas  on 
the  official  plat.  This  is  sometimes  difficult,  the  areas  in 
some  cases  seeming  to  have  been  put  down  without  any 
calculation. 


fractional   by 

FIG.  70. 


waters,   reser- 


Sections  made 

vations,  etc., 
should  be  sub- 
divided in  such 
a  manner  as  to 
produce  the 
same  result  as 
would  have 
been  produced 
had  the  section 
been  full.  This 
may  sometimes 
be  done  by  ex- 
tending and  by 
measuring  the 
lines  on  the  ice, 
or  over  the  res- 
ervation. 


figure  illustrating  the  Subdivision  of  a  Section 
fractional  on  waters. 

Commissioner  Drummond  says  (see  Copp's  Land  Laws, 
p.  761):  "In  the  subdivision  of  fractional  sections,  where  no 
opposite  corners  have  been  or  can  be  fixed,  the  subdivision 
lines  should  be  ascertained  by  running  lines  from  the  es- 
tablished corners  due  north,  south,  east  or  west,  as  the  case 


SUBDIVISION  OF  SECTIONS.  273 

may  be,  to  the  water-course,  Indian  boundary  line,  or 
other  external  boundary  line  of  such  fractional  section. 
The  law  presupposes  the  section  lines  surveyed  and 
marked  in  the  field  by  the  United  States  deputy  survey- 
ors to  be  due  north  and  south  or  east  and  west  lines. 
But  in  actual  experience,  this  is  not  always  the  case. 
Hence,  in  order  to  carry  out  the  spirit  of  the  law,  it  will 
be  necessary  in  the  running  of  subdi visional  lines  through 
fractional  sections  to  adopt  mean  courses  where  the  lines 
are  not  due  lines,  or  to  run  the  subdivisional  line  paral- 
lel with  the  section  line  when  there  is  no  opposite  section 
line.'1 

4.  Irregular  Subdivisions  of  Fractional  Sec- 
tions.—In  making  irregular  subdivisions  of  fractions 
bounded  on  streams  or  lakes,  the  following  rule  has 
been  laid  down  by  the  authorities. 

It  has  been  decided  by  the  Supreme  Court  of  the  United 
States  that  "the  meander  lines  run  in  surveying  frac- 
tional portions  of  the  public  lands  bordering  upon  navi- 
gable rivers  are  run  not  as  boundaries  of  the  tract  but 
for  the  purpose  of  defining  the  sinuosities  of  the  stream 
and  as  the  means  of  ascertaining  the  quantity  of  land  in 
the  fraction,  and  which  is  to  be  paid  for  by  the  pur- 
chaser." 

R.  R.  Co.  v,  Schurmier,  7th  Wallace  (U.S.)  272. 
It  has  been,  held  that  the  same  lines  are  to  be  used  in 
ascertaining  the  quantity  of  land  in  any  portion  of  the 
fraction.  Thus,  as  often  happens,  if  a  deed  calls  for  so 
many  acres  off  the  end  of  the  fraction,  the  surveyor  in 
making  his  computations  to  determine  at  what  point  to 
locate  the  dividing  line,  should  in  the  absence  of  any- 
thing showing  to*  the  contrary,  use  the  meander  line  for 
the  purpose  of  estimating  the  area  of  the  tract,  and  lay 
down  the  dividing  line  accordingly.  Otherwise  there 
could  be  no  common  basis  of  calculation  and  as  many 
different  results  would  be  arrived  at  as  there  were  differ- 
ent surveyors  to  run  the  line,  or  different  times  of  survey. 


274  A  MANUAL  OF  LAND  SURVEYING. 

This  is  especially  true  of  fractions  bordering-  on  lakes 
whose  shore  lines  are  subject  to  great  change  from  natu- 
ral causes  or  artificial  drainage. 

The  common  law  rule  for  calculating  the  quantity  of 
land  bordering  on  a  non-navigable  stream  is  that  no  ref- 
erence is  had  to  what  lies  between  low  water  mark  and 
the  centre  of  the  stream.  On  navigable  waters,  high 
water  mark  is  the  line. 

Lamb  v.  Eickett,  11  Ohio  311. 

5.  Exceptional  Oases.— In  the  United  States  sur- 
veys made  previous  to  1815,  there  was  much  irregularity 
in  the  practice  of  the  surveyors  in  carrying  on  the  sur- 
veys. The  fractional  sections  were  frequently  thrown 
upon  the  south  or  east  tiers  of  sections  in  the  township; 
the  surveys  being  carried  on  from  the  north  to  the  south 
and  from  the  west  to  the  east.  Where  the  township  was 
made  fractional  by  large  rivers  or  lakes,  they  were  fre- 
quently so  laid  off  as  to  throw  all  the  fractions  into  the 
sections  bordering  on  the  water. 

There  was  even  greater  irregularity  in  the  manner  of 
subdividing  the  fractional  sections  into  the  lesser  tracts. 
Many  of  them  had  no  quarter  section  corners.  In  some, 
the  government  plats  show  no  subdivision;  some  are  sub- 
divided in  one  way  and  some  in  another. 

In  making  resurveys  and  subdivisions  of  these  and  all 
other  exceptional  cases,  the  surveyor  must  always  make 
his  resurvey  conform  to  the  plan  as  shown  by  the  field- 
notes  and  plats  of  the  original  survey. 

6.  Field  Notes  of  the  Survey  and  Subdivi- 
sion of  a  Fractional  Section.  — The  following 
notes  of  an  actual  survey  are  intended  as  an  illus- 
tration of  the  manner  of  subdividing  a  section 
under  the  ordinary  conditions  as  they  are  met  with 
in  the  field,  including  the  manner  of  restoring, 
certain  lost  corners  and  to  a  certain  extent  the 
principles  governing  re-surveys  as  laid  down  in 


SUBDIVISION    OF    SECTIONS. 


275 


subsequent  chapters.  All  the  corners  of  the  United 
States  survey  have  equal  weight  or  authority,  hence 
in  subdividing  a  section  or  restoring  lost  corners  of 
that  survey  it  makes  no  difference  at  what  corner 
the  surveyor  begins  his  work  or  in  what  order  it  is 
done,  except  so  far  as  his  own  convenience  and  that 
of  his  assistants  are  concerned.  It  will  be  rarely,  if 
ever,  that  the  work  will  be  done  in  precisely  the 
same  order  in  any  two  sections  in  a  settled  country. 
The  student  should  trace  the  notes  of  each  operation 
carefully  through  and  verify  the  results.  The  first- 
thing  required  by  the  surveyor  is  a  complete  and 
correct  copy  of  all  the  field  notes  of  record  which 
refer  to  the  section  to  be  subdivided  and  of  adjacent 
sections  when  necessary  for  restoring  lost  corners. 
In  order  that  the  student  may  be  able  to  trace 
through  the  several  operations  properly  and  under - 
standingly  so  much  of  the  field  notes  as  were  re- 
quired in  the  survey  are  given  herewith. 

Field  Notes  of  U.  S.  Survey  of  Section  2  T4  S, 
E  9  W.  North  Boundary.     Var.  5°  10 '  E. 


West 
71.78 
80.00 


West 
22.50 
39.00 
40.00 


59.59 
66.44 
69.22 

80.00 


On  S.  boundary  of  sec.  36,  T  3  S,  R  9  W. 

Black  Ash  16  in. 

Set  post  cor.  to  sees.  35  and  36. 

Beech  10  in.,  N.  30  W.  10  Iks. 

Tam'k  6  in.,  N.  46^  E.  78  Iks. 


On  S.  boundary  of  sec.  35,  T  3  S,  R  9  W. 

Lynn  6  in. 

Left  Swamp. 

Set  post  X  sec.  cor.  sec.  35. 

Beech  9  in.,  N  39  W  39  Iks. 
Beech  6  in.,  N  74  E  48  Iks. 
Sugar  14  in. 

Stream  20  Iks.  wide,  course  south. 
Stream  10  Iks.  wide,  course  south. 
Set  post  cor.  sees.  34  and  35. 

Beech  6  in.,  N.  58^  W.  33  Iks. 

Beech  6  in.,  N.  52  E.  31  Iks. 

John  Mullett  D.  S.,  Nov.  8,  1825. 


2*76 


A    MANUAL    OF    LAND    SURVEYING. 


West 
39.95^ 


Subdivisions.       Var.   5°  35'  E. 

Between  sections  11  and  12. 
Stream  25  Iks.  wide,  course  S.  E. 
Set  post  X  sec-  cor-  Sees.  11  and  12. 

Water  Beech  6  in.N.  55  E.  11  Iks. 

Sycamore  30  in.  N.  77  W.  28^  Iks. 
Set  Post  corner  to  Sees.  1,  2,  11,  and  12. 

Tam'k  6  in.  S.  18  W.  37  Iks. 

Tam'k  8  in.  N.  88  E.  21  Iks. 


Corrected  line  between  Sees.  1  and  12. 
Set  X  Sec.  post. 

Beech  16  in.  N.  23  E.  31  Iks. 

Beech    8  in.  S.  41  W.  17>^  Iks. 
Sec.  Cor. 


Between  Sees.  1  and  2. 

Elm  HO  in. 

Set  post  ^  sec.  cor.  Sees.  1  and  2. 

Beech  13  in.  S.  74  E.  26  Iks. 

Beech  6  in.  N.  30^  W.  11  Iks. 
Entered  swamp. 

Intersected  N.  boundary  20  Iks.  E.  of  post.     Set 
post  at  intersection.     Cor.  to  Sees.  I  and  2. 

Black  ash  16  in.  S.  65  W.  33  Y2  Iks. 

Black. ash  15  in.  S.  47  E.  39  Iks. 


Between  Sees.  10  and  11. 

Set  post  #  sec.  cor  Sees.  10  and  11. 

Beech  8  in,  S.  78  E.  34  Iks. 

Beech  10  in.  S.  79  W.  13^  Iks 
Set  post  cor.  to  Sees.  2,  3,  10,  and  11. 

Beech  7  in.  S.  62  E.  25^  Iks. 

Beech  14  in.  N.  63  W.  3  Iks. 


On  random  between  Sees.  2  and  11. 
Enter  tam'k  and  birch  swamp. 
Stream  25  Iks.  wide,  course  S.  E. 
Intersected  E.  boundary  25  Iks.  S.  of  post. 

"Corrected  line  between  Sees.  2  and  11. 
Set  quarter  Sec.  post. 

Beech  12  in.  S  50  E.  3  Iks. 

Beech  8  in.  N.  21  W.  18  Iks. 
Sec.  Cor. 

Between  Sees,  2  and  3. 
Entered  swamp. 

Brook  10  Iks.  wide,  course  S.  W. 
Left  Swamp. 


40.00 


61.48 
78-32 


SUBDIVISION    OF    SECTIONS.  277 

Set  post  ^  sec.  cor.  Sees.  2  and  3. 

Whitewood  20  in.  N.  53  W.  27  Iks. 
Beech  8  in.  N.  89  E.  20  Iks. 
Beech  12  in. 

Intersected  N.  boundary  45  Iks.  E.  of  post.     Set 
post  at  intersection.     Cor.  to  Sees.  2  and  3. 
Beech  8  in.  S.  74  \V.  33  Iks. 
Beech  14  in.  S.  16  E.  47  Iks. 

Robert  Clark,  Jr.,  D.  S. 
May  6,  1826. 


Notes  of  Later  Survey. 

On  May  12,  1854,  Randolph  Nutting  found 
the  east  and  the  west  quarter  section  corners  of  this 
section,  with  bearing  trees  standing  and  ran  the 
east  and  west  quarter  line  between  them,  setting 
temporary  stakes  on  the  true  line,  dividing  it  into 
four  equal  parts.  He  also  renewed  the  posts  at  these 
quarter  section  corners.  At  the  corner  of  Sections 
2,  3,  10,  and  11  he  found  the  decayed  post  and  bear- 
ing tree/  Beech  14  N.  63  W.  3  Iks.,  standing,  the 
other  bearing  tree  destroyed.  He  planted  a  granite 
boulder  12x16x24  in.  one  foot  below  the  surface 
and  at  the  corner  drilled  a  hole  l£  in.  diameter 
and  4  in.  deep.  Thence  he  ran  east  on  random 
40.12  ch.  to  the  quarter  section  corner  and  correct- 
ing back  at  20.06  ch.  on  true  line  set  granite  boulder 
10x12x18  in.  1  foot  beneath  the  surface,  marked 
it  with  a  drill  hole  and  planted  side  stones  50  links 
each  north  and  south  from  the  corner. 

Re- Survey  and  subdivision  of  Section  2,  T  4  S,  E  9, 
W.,  April  4,  5,  6,  1882. 

(Note :  For  convenience  of  reference  the  corners 
of  the  several  subdivisions  are  numbered  or  lettered 
as  shown  in  the  figure,  on  the  system  shown  at  the 
close  of  Chapter  VI. ) 


278  A    MANUAL    OF    LAND    SURVEYING. 

5 


14  7 

PLAT  OF  SEC  2. 

r 

Began  at  4,  where  I  found  the  "rock  bound" 
planted  by  Nutting,  18  in.  below  surface  at  intersec- 
tion of  roads.  All  parties  agree  that  this  corner 
stone  has  not  been  moved  from  its  original  position. 
All  traces  of  bearing  trees  are  destroyed  or  removed. 
I  accept  the  corner  as  correct  and  mark 

Sugar  12  in.  N.  67  E.  68  Iks. 

Beech  9  in.  N.  37  W.  76  Iks. 


,40.15 


Ran  thence  north  on  random,  setting  stakes 
every  10  chains. 

Intersected  the  quarter  line  7  Iks.  east  of  8.  Cor- 
ner post  dug  out  in  the  road.  The  stump  of  white- 
wood  bearing  tree  is  standing  and  I  ran  thence  S. 
53  E.  27  Iks.  and  plant  for  X  Sec.  Corner  an  earth- 
enware post  3  in.  diam.  30  in.  long  with  stones 
around  it  and  mark 

Beech  8  in.  N.  26  E.  76  Iks. 
Sugar  10  in.  S.  47  E.  104  Iks. 


SUBDIVISION    OF    SECTIONS.  279 


— 

South  corrected  line. 

Set  granite  boulder  8  x  12  x  24  in.  marked  -f  for 
corner  (15)  with  side  stones  50  Iks.  each  east  and 
west.  Put  a  back  sight  on  this  corner  and  return  to  8. 
Set  up  transit  over  the  post,  back  sight  to  15  and 
prolong  line  north,  setting  temporary  stakes  at 
20.00  chs.  and  38.32  chs.  and  search  for  sec. 
corner.  Bearing  trees  are  both  destroyed  and 
obliterated  in  the  jog  in  the  road.  Chapm  says  a 
new  stake  has  been  driven  in  the  old  stake  hole, 
and  points  out  the  location,  but  on  digging  I  find 
nothing  of  it  there.  I  then  look  for  the  corner  of 
Sees.  34  and  35.  I  find  remains  of  a  stump,  which 
from  its  position,  may  have  belonged  to  one  of  the 
bearing  trees.  I  set  transit  up  over  it  and  run 
S.  58  >£  E.  33  Iks.  and  dig  the  earth  carefully  away. 
I  find  the  decayed  remains  of  a  stake  from  which 
point  I  run  east  45  links  and  dig  and  find  the  new 
stake  driven  in  the  old  stake  hole,  as  described  by 
Chapin.  Random  line  from  the  south  intersects 
the  township  boundary  2  Iks.  east  of  the  corner  of 
Sees.  2  and  3  at  38.46  chains  from  the  quarter  post. 
I  plant  new  stake  in  old  stake  hole,  cor.  of  Sees.  34 
and  35  and  for  cor.  of  Sees.  2  and  3  plant  iron 
landside  of  plow  packed  about  with  brickbats  and 
broken  crockery  and  mark 

Sugar  10  in.  S.  43*4  W.  76  Iks. 
Sugar  14  in.  S.  32  E.  1.24  Iks. 


Thence  south  corrected  line. 
Set  stake  with  broken  brick  and  glass  around 
for  corner  (16)  148  Iks.  south  of  stump  of  Beech  line 
tree; 

Cherry  18  in.  N.  72  E.  64  Iks. 
Elm  24  in.  S.  61^  E.  96  Iks. 


I  then  return  to  8  and  offset  north  20  itis.  and 
set  up  transit  in  the  section  line.  Backsight  to  (15) 
and  run  thence  east  at  right  .angles  with  section 
line,  setting  stakes  every  10  chs.  on  random  line. 

To  bank  of  Mill-pond.  Set  flag  in  line  across 
pond  and  then  run  a  line  south  at  right  angles  with 
random  5.00  eh.  to  a  point  where  I  set  up  transit 
and  measure  the  angle  between  lines  to  back-sight 
and  to  flag  over  the  pond=54°  26 '  whence  nat.  tan, 
1.3985X5.00=6.9925  makes  the  distance  across 
pond  6.99^  and  distance  on  random  over  pond= 
43.00-j-6.99#=49.99i4:.  Continue  the  line. 


280 


A    MANUAL    OF    LAND    SURVEYING. 


80.00 


118.00 


80.00 


38.12 


Set  temporary  stake  and  look  for  #.  sec.  cor. 
Sees.  1  and  2.  There  are  no  bearing' ^tfrees  stand- 
ing and  no  one  knows  the  location  of  the  corner. 
I  set  stakes  to  mark  the  random  line  and  go  next 
to  3,  where  I  find  no  traces  or  evidence  ot  the  cor- 
ner. I  then  go  to  the  #  sec.  cor.  of  Sees.  11  and 
12,  where  I  find  both  bearing  trees  and  corner  post 
standing  in  place. 


From  thence  I  run  north  on  random,  setting  temp, 
stakes  at  40.00,  60.00  and  80.00  chains." 

Set  temp,  stake  and  look  for  corner. 

The  bearing  trees  of  both  corners  are  missing, 
but  there  are  numerous  stumps  near  by  rotted  to 
the  ground.  I  go  east  along  township  boundary 
and  find  blk  ash  station  tree  standing.  I  take  its 
distance  fromTp.  cor.  at  71.78  chains  and  continue 
the  measure  west  on  random. 

Set  temp,  stake  and  search  again,  and  finding 
what  appears  likely  to  be  the  remains  of  stump  of 
bearing  tree,  run  thence  N.  47  W.  39  Iks.  stick  a 
pin  and  run  from  it  S.  65  W.  33^  Iks.  to  roots  of 
black  ash  tree  lying  on  the  ground  decayed  and 
moss  covered.  Digging  at  the  point  where  I  stuck 
the  pin,  I  find  18  in.  below  the  surface  in  wet  soil, 
the  sound  bottom  of  the  original  post,  cor.  of  Sees. 
1  and  2.  As  a  check  I  measure  west  20  Iks.  and  by 
digging  find  the  post  cor.  of  Sees.  35  and  36  and 
plant  a  new  post  in  its  place.  At  the  cor.  of  Sees. 
1  and  2  I  put  in  a  piece  of  iron  plow  beam  24 
inches  long,  pack  brick  and  stone  about  it  and  set 
side  stones  50  Iks.  each  east  and  south  from  corner. 
Random  line  from  the  south  at  118.39  ch.  inter- 
sects the  township  boundary  12  Iks.  west  of  corner 
of  Sees.  1  and  2. 


South  between  Sees.  1  and  2. 

Find  by  digging  the  remains  of  decayed  post  in 
position  9  Iks.  east  of  random  line.  ;  I  also  find 
roots  of  bearing  trees  at  corresponding  points 
called  for  by  field  notes. 

Random  line  from  the  west  at  80.04  ch.  inter- 
sects E.  bound  45  Iks.  north  from  qr.  sec.  cor. 

I  plant  a  stone  5  x  8  x  16  marked  4-  with  brick- 
bats and  broken  glass  around  for  14 
mark 

Beech  12  in.  E.  63  Iks., 
'  Beech  14  in.  S.  29  W.  92  Iks. 


sec.  cor.  and 


SUBDIVISION"    OF    SECTIONS.  281 


Set  flag  in  line  at  6  and  continue  thence  south. 
Set  temp,  stake  6  Iks.  east  of  random  line.  Search 
the  ground  carefully  and  find  no  traces  of  corner 
stake  or  of  bearing  trees. 

I  next  return  to  4,  set  up  flag  on  the  corner  and 
measure  thence  east  along  the  highway. 

Corner  14  "Rock  bounds"  set  by  Surveyor 
Nutting. 

Set  up  transit  at  corner  14,  backsight  to  4  and 
prolong  the  line  east. 

Original  corner  and  bearing  trees  entirely  dug 
out  and  destroyed  in  the  highway.  Punch  a  hole 
with  on  iron  bar  4  ft.  deep  and  three  in.  diameter 
and  fill  it  with  Portland  cement  mortar  for  quarter 
section  cor.  and  mark  maple  10  N.  5  W.  62  Iks. 

Set  temp,  stake  and  continue  measure  east  with- 
out running  the  line,  the  qr.  post  of  Sees.  1  and  12 
being  known. 

Find  corner  post  and  bearing  trees  standing  at 
quarter  section  Cor.  See's  1  and  12. 

I  then  return  to  3  and  locate  section  corner  at 
distances  from  nearest  corners  N.  S.  E.  and  W.  as 
follows: 

From  quarter  sec.  cor.  Sees.  1  and  2  40.13  chs. 
From  quarter  sec.  cor.  Sees.  11  and  12  40.13  chs. 
From  quarter  sec.  cor.  Sees.  1  and  12  40.05^  chs. 
From  quarter  sec.  cor.  Sees.  2  and  11  40.14  chs. 

I  drive  a  black  walnut  stake  deep  in  the  ground 
at  the  corner  and  over  it  put  a  stone  8  x  12  x  30  in. 
marked  -f- 

Black  Ash  8  in.  S.  43  W.  82  Iks., 
Black  Ash  10  in.  N.  51  W.  126  Iks. 

Thence  north  on  true  line  (Var.  of  needle  2°  34' 
E.) 

Set  stake  with  brick  around  for  corner  12  and 
mark 

Elm  36  in.  N.  87  W.  54  Iks. 
Cherry  24  in.  S.  5  W.  68  Iks. 


I  then  return  to  7,  set  up  transit  over  the  corner, 
sight  to  flag  at  14  and  turn  angle  to  right  of  89° 
40'  and  run  thence  at  that  angle  north  on  random 
setting  temporary  stakes  every  10  eh. 

Intersected  E.  and  W.  corrected  quarter  sec.  line. 

To  point  on  bank  of  mill  pond.  Set  two  flags 
in  Hue  over  pond  to  range  by,  then  angle  to  the 
left  60°  and  run  5.00  to  2nd  triangulating  point 


282 


A    MANUAL    OF    LAND    SURVEYING. 


60.00 

78.42 


18.21^ 


20.01 


40.13 


20.06^ 


from  which  I  turn  off  angle  of  90°  to  right  and  set 
flag  in  the  random  line  over  the  pond.  Distance 
between  the  triangulating  points  1  and  3=5.00  X 
secant  60°  (2. 0000) =10. 00  ch. 

Point  in  random  over  pond. 

Intersect  N.  boundary  36  Iks.  east  of  quarter  sec- 
tion cor.  Sec.  35.  Found  both  bearing  trees  stand- 
ing for  quarter  section  cor.  Sec.  35  and  decayed 
stake  in  right  place  for  corner.  Planted  a  new 
stake  with  stones  around  and  ran  thence  east  32j^ 
Iks.  and  planted  a  piece  of  ll/2  iron  pipe  3  ft.  long 
for  quarter  section  corner  Sec.  2  and  put  brick 
around  it  and  marked 

Birch  10  in.  S.  48  E.  116  Iks. 
Beech  12  in.  S.  16  E.  32  Iks. 


South  corrected  quarter  section  line. 
Set  stake  with  glass  and  broken  crockery  around 
for  corner  and  marked 

Swamp  oak  12  in.  S.  2  E.  186  Iks. 
At  intersection  of  quarter  section  lines  set  stone 
9  x  13  x  24  with  stake  underneath  and  cross  on  top 
at  corner. 

Wh.  ash  8  in.  N.  12  W.  10  Iks. 
Hickory  12  in.  N.  84  E.  63  Iks. 
Intersection  is  40.02  ch.  from  8. 


From  C  west  corrected  line. 

Set  stake  with  brick  and  glass  around  for  cor- 
ner d.  No  trees  near. 

From  d  south  on  random  at  angle  .of  90°  with 
random  E.  and  W.  ^  line. 

Intersected  Sec.  line  4  Iks.  west  of  corner  14. 


North  corrected  line. 

Set.  stone  6  x  8  x  16  marked   -j-  for  cor.  h.    No 
tree  near. 

Returned  to  c  and  on  corrected  line  20.07  from 
C  and  7  set  post  with  broken  glass  around. 
Cherry  10  in.  S.  67^  E.  93  Iks. 
Whitewood  16  in.  S.  21  W.  114  Iks. 


OTHER    ORIGINAL    SURVEYS.  283 

This  survey  was  made  with  transit  and  steel  tape. 
On  all  lines  run  with  the  transit  temporary  stakes 
were  set  every  ten  chains  on  the  random  line  and 
afterward  corrected  to  the  true  line.  Let  the  stu- 
dent find  the  amount  and  direction  of  the  correction 
applied  to  each  stake  to  place  it  in  the  true  line. 
The  measurements  on  this  section  were  more  than 
usually  uniform  on  different  lines,  the  ground  being 
comparatively  level.  Usually  the  quarter  section 
corners  set  in  the  old  compass  surveys  are  more  or 
less  out  of  line  between  the  section  corners,  causing 
a  discrepancy  in  the  interior  measurements.  Differ- 
ence in  the  character  of  the  surface  of  the  land 
along  different  lines  also  has  a  tendency  to  cause 
discrepancy  of  measures.  It  will  be  also  noticed 
that  in  this  survey  the  direction  of  the  lines  is  con- 
trolled by  the  monuments  of  the  original  survey 
and  their  absolute  direction  from  the  true  meridian 
was  considered  of  so  little  importance  that  no  atten- 
tion was  paid  to  it.  In  retracing  the  more  careful 
surveys  now  being  made  by  the  United  States  sur- 
veyors, the  direction  from  the  true  meridian  is  a 
more  important  factor  in  the  resurvey. 

7.  Other  Original  Surveys. — In  a  considerable 
portion  of  the  United  States,  the  general  government 
never  had  any  ownership  of  the  land. 

The  surveys  were  there  made  by  the  proprietors  upon 
such  system  or  plan  as  suited  themselves. 

The  further  subdivision  of  these  tracts  is  original  sur- 
veying. It  is  sufficient  to  say  of  this  work  that  it  should 
be  done  with  great  care,  and  that  ihe  marks  upon  the 
ground  which  indicate  the  boundary  lines  should  be  <tf 


284  A    MANUAL    OF    LAND    SURVEYING. 

the  plainest  and  most  permanent  character  which  th* 
circumstances  of  the  case  permit, 

These  marks  are  intended  to  fix  for  all  time  the  boun- 
daries  of  the  tract  laid  off  and  they  cannot  be  too  plain  or 
permanent.  Want  of  due  care  and  precaution  in  making" 
permanent  land  marks  upon  the  ground,  at  the  time  of 
the  original  survey,  is  the  fruitful  cause  from  whicn 
arises  most  of  the  litigation  about  boundary  lines. 

8.  Highway  surveys,  like  other  surveys,  lose  much 
of  their  value  if  their  corners  and  lines  are  not  so  thor- 
oughly marked  as  to  be  readily  found  at  any  future  time. 
The  centre  line  of  a  highway  is  very  commonly  used  as  a 
boundary  line.    Good  permanent  landmarks,  well  guarded 
by  bearings  and  distances  to  the  most  permanent  objects 
in  the  vicinity   should  be  planted  at  the  starting  and 
closing  points  of  the  survey,  at  each  angle  in  its  course, 
and  at  every  crossing  of  a  section  line.    The  distance  of 
the  crossing  points  should  be  given  from  the  nearest  gov- 
ernment corners  each  way  on  the  section  line. 

9.  Surveys  for  town  plats  are  made  upon   any 
system  to  suit  the  circumstances  of  the  case,  or  the  views 
of  the  owners  of  the  land  platted. 

In  making  these  surveys,  it  is  important  that  the  work 
be  in  every  respect  carefully  done;  that  full  and  complete 
notes  bo  taken,  so  that  the  plat  when  finished  shall  show 
every  material  fact  which  may  be  of  use  to  the  public  or 
to  the  future  surveyor. 

The  relation  which  the  lines  of  the  plat  bear  to  the 
lines  of  the  original  boundaries,  whether  of  the  govern- 
ment survey  or  otherwise,  should  be  shown  on  the  plat, 
and,  what  is  most  important  of  all,  the  location  of  the 
lines  upon  the  ground  should  be  marked  by  a  sufficient 
number  of  permanent  monuments  so  that  there  may  never 
arise  any  difficulty  in  determining  the  exact  position 
those  lines  occupy. 

Such  monuments  should  be  placed  at  the  corners  and 
angles  of  the  tract  platted,  and  if  included  in  the  United 


OTHER    ORIGINAL    SURVEYS.  285 

States  survey,  they  should  be  placed  at  the  corners  of  the 
legal  subdivisions  of  a  section  which  are  included  in  the 
plat.  Monuments  should  also  be  placed  so  as  to  define 
the  lines  and  termini  of  all  streets. 

For  this  purpose,  they  may  be  placed  either  along  the 
centre  lines  and  angles  of  the  streets  or  along  their  mar- 
gins at  the  corners  and  angles  of  blocks.  Each  method 
has  its  advantages  and  disadvantages.  The  surveyor 
should  consider  the  special  circumstances  of  each  case, 
and  so  locate  the  monuments  that,  while  effecting  the 
purpose  for  ^which  they  were  intended,  they  shall  be 
the  most  likely  to  remain  in  position  and  the  easiest  to 
refer  to. 

10.  Tu  Michigan  town  plats  are  required  by  law  (Session  Laws  of 
1885)  to  be  made  and  recorded  in  the  following  manner: 

The  plats  must  be  made  on  sheets  of  good  muslin  backed  paper,  18 
inches  by  24  inches  in  size,  on  a  scale  showing  not  more  than  200  feet 
to  an  inch. 

The  plat  must  have  upon  it  a  full,  detailed  written  description  of  the 
land  embraced  in  it,  showing  the  township,  range,  section  and  subdi- 
vision of  section  of  the  land  platted.  If  the  premises  platted  are  not 
included  in  the  legal  subdivisions  of  the  government  survey,  then  the 
boundaries  are  to  be  denned  by  metes  and  bounds  and  courses. 

The  plat  must  contain  the  full  name  of  the  town,  city,  village  or  ad- 
dition platted;  the  names  of  the  proprietors  and  of  the  person  making 
the  plat,  and  the  date. 

It  must  be  signed  by  the  proprietors  and  by  the  person  making  it, 
and  be  witnessed  and  acknowledged  in  the  same  manner  as  deeds. 

The  sections  and  parts  of  sections  must  also  be  designated  on  the 
plat  by  lines  with  appropriate  letters  and  figures. 

There  must  be  a  plain  designation  of  the  cardinal  points  of  the  com- 
pass and  a  correct  scale. 

When  complete  and  before  any  copies  are  made  from  it,  the  plat 
must  be  submitted  to  the  Auditor  General  for  his  approval, 

11.  The  Record.— An  exact  duplicate  of  the  original  plat  must 
be  filed  in  the  office  of  the  Register  of  Deeds  for  the  county  in  wtych 
the  land  is  situated.    It  must  contain  all  the  matter  in  the  original 
plat  and  the  certificate  of  the  Register  of  Deeds  and  the  person  who 
made  the  original  plat,  that  they  have  separately  carefully  compared 
the  duplicate  with  the  original  plat,  and  that  it  is  an  exact  duplicate 
thereof  and  of  the  whole  of  such  plat. 


286  A    MANUAL    OF    LAND    SURVEYING. 

A  third  copy  must  be  filed  in  the  office  of  the  Auditor  General. 
This  copy  must  contain  the  certificate  of  the  Register  of  Deeds  and  of 
the  person  who  made  the  plat,  that  they  have  separately  compared  it 
with  the  duplicate  plat  on  record,  and  that  it  is  a  true  transcript  ther^e- 
from  and  of  the  whole  of  such  duplicate  plat  so  recorded. 

The  Register  of  Deeds  receives  a  fee  of  $2.00  for  recording  the  plat, 
and  the  sum  of  $1.00  must  accompany  the  plat  filed  in  the  Auditor 
General's  office. 

The  law  was  amended  in  1887  so  as  to  require  the  surveyor  to 
plant  permanent  monuments  at  all  angles  in  the  boundaries  of 
the  land  platted,  and  at  all  the  intersections  of  streets,  or  streets 
and  alleys,  as  shown  on  the  plat ;  and  when  there  are  permanent 
objects  in  the  vicinity  of  such  monuments,  the  bearings  and  dis- 
tances of  such  objects  to  be  noted.  The  character  of  the  monu- 
ments and  the  bearings  and  distances  of  such  objects  or  witness 
points  must  be  given  in  the  most  convenient  manner  on  the  plat. 
The  surveyor  must  certify  that  the  plat  is  a  correct  one,  and  that 
the  monuments  described  in  it  have  been  planted  as  therein 
described.  The  new  provisions  of  the  law  are  very  important. 
The  monuments  are  the  crowning  work  of  the  survey,  without 
which  all  else  is  of  little  value.  They  mark  out  the  standard  of 
measure  on  the  grouud,  to  which  all  subsequent  surveys  must 
conform.  President  Steele,  of  the  Michigan  Engineering  Society, 
says :  "  Place  more  monuments  instead  of  less.  Place  them 
everywhere,  no  matter  whether  at  the  intersections  of  streets  at 
the  side  lines,  the  centre  lines,  or  any  other  lines.  Put  down  all 
you  can.  Plant  them  in  exact  relation  one  to  another.  Put  the 
bearing  on  every  line,  the  angle  at  every  intersection.  Put  it  all 
on  your  plat,  and  the  more  you  have  the  better.  Leave  nothing 
to  guess  at.  Have  it  so  plain  that  a  man  who  never  knew  any- 
thing about  the  ground  can  go  there  and  find  all  the  points." 
17 


MONUMENTS.  28*7 

IT .  Monuments.— It  is  more  important  to  a  man 
to  know  precisely  where  his  boundary  lines  are  and  that 
they  are  unchangeable  without  his  consent,  than  it  is  that 
he  shall  have  the  precise  quantity  of  land  ;  hence  one  of 
the  most  important  duties  the  surveyor  has  to  perform,  is 
to  fix  the  most  permanent  and  unmistakable  monuments 
to  define  and  preserve  boundary  lines.  This  is  equally 
true  of  all  original  surveys,  whether  in  country  or  town. 
Mathematical  accuracy  in  measuring  distances  or  running 
lines,  fails  of  its  purpose  unless  there  be  some  means  of 
securing  an  unvarying  starting  point;  while  if  the  land- 
marks of  the  original  survey,  in  accordance  with  which 
the  land  was  conveyed,  be  preserved  intact,  no  measure- 
ments, good  or  bad,  are  needed  to  define  the  boundaries. 

Monuments  for  landmarks  should  be  durable  and  easily 
distinguishable  from  other  objects  in  the  vicinity. 

They  should  be  accessible,  not  liable  to  be  moved,  and 
their  position  located  by  bearings  and  distances  to  the 
most  permanent  objects  in  the  vicinity. 

Various  things  are  used  for  landmarks — according  to 
the  nature  of  the  soil  and  the  materials  at  hand  ;  chiefly 
wood,  stone,  earthenware,  or  iron,  in  some  of  their  forms. 

Wood.  A  wooden  post,  if  of  suitable  size  and  kind  and 
properly  planted,  makes  an  excellent  landmark,  where 
very  precise  definition  of  the  boundary  is  not  required. 
It  should  be  from  2%  to  4  inches  in  diameter,  sound  and 
straight  and  planted  vertically  in  the  ground  to  a  depth 
of  at  least  three  feet,  for  permanent  purposes.  Ked  cedar 
black-walnut,  cherry  or  white  oak  hearts  make  very  dur- 
able posts.  When  the  post  has  decayed  the  rotten  wood 
and  cavity  in  the  earth  preserve  the  point  better  than  the 
sound  post,  as  they  cannot  be  pulled  up  nor  moved  from 
place  without  moving  the  surrounding  earth  with  them. 

Stone.  If  a  rough  stone  or  boulder  is  used  for  a  monu- 
ment, it  should  either  be  so  large  as  not  to  be  moved  by 
any  ordinary  accident  or  so  firmly  imbedded  in  the  earth 
as  to  defy  the  plow  or  the  road  maker.  If  of  a  kind  com- 
mon in  the  vicinity,  it  should  be  very  plainly  marked  and 
have  some  foreign  material  like  brick,  iron,  glass,  ot 
crockery  imbedded  around  it,  to  identify  it  by. 


288  A    MANUAL    OF    LAND    SURVEYING. 

If  cut  stone  is  used,  it  should  be  of  the  best  quality  and 
be  long  enough  and  set  deeply  enough  to  insure  perma- 
nency. If  the  stone  is  a  soft  one  it  should  be  protected 
from  injury.  A  stone  36x8x8  inches  dressed  down  at  the 
top  to  6x6  inches  is  the  size  in  use  in  many  of  the  large 
cities  for  landmarks.  It  is  common  to  cut  a  cross  or  drill 
a  hole  in  the  top  of  the  stone  to  indicate  more  precisely 
the  corner  or  line.  If  still  greater  precision  is  required  a 
piece  of  metal  is  set  in  the  stone  and  the  point  indicated 
by  lines  cut  as  finely  as  desirable. 

Iron.  Monuments  of  cast  iron  have  been  used  and  are 
excellent.  A  hollow  cone  18  to  36  inches  in  length  with  a 
broad  flange  at  the  bottom,  when  set  in  the  ground  holds 
its  position  very  firmly  and  will  last  indefinitely.  Iron 
rods,  and  pieces  of  gas  pipe  are  also  used.  They  need  to 
be  well  packed  about  the  top  with  brick  or  stone  to  keep 
them  in  position. 

Other  Materials.  Some  monuments  are  made  of  the 
same  material  as  the  earthenware  sewer  pipe,  and  burned 
and  glazed  in  a  similar  manner.  They  are  solid,  cylindri- 
cal, three  inches  in  diameter  and  of  various  lengths.  The 
ends  are  suitably  marked  before  burning.  They  are 
very  convenient  to  use  and  durable,  but  need  to  be  well 
protected.  Brick  set  on  end  two  and  two  to  a  depth 
of  three  feet  and  packed  about  the  top  to  prevent 
moving  make  an  excellent  monument.  Another  excel- 
lent device  is  to  make  a  deep  hole  in  the  earth,  one  or  two 
inches  in  diameter,  and  fill  it  with  a  paste  of  quick  lime, 
plaster  of  paris,  or  portland  cement. 

Protection.  A  good  plan  for  protecting  monuments  in 
the  streets  of  a  town,  is  to  place  them  in  shallow  pits  a 
foot  or  more  in  diameter.  Set  the  monument  in  the  pit 
so  that  the  top  of  it  shall  be  several  inches  above  the  bot- 
tom of  the  pit  and  as  much  below  the  street  pavement. 
Protect  it  with  a  cast  iron  cylinder  set  about  it,  having  a 
slightly  conical  cover  which  is  level  with  and  forms  part 
of  the  pavement.  The  summit  of  the  cover  answers  some 
of  the  purposes  of  the  monument,  while  by  removing  it 
the  monument  Itself  is  brought  to  view. 


RESURVEYS.  289 

CHAPTER  X. 

RESURVEYS. 

1.  In  an  old  settled  country,  the  principal  work  of 
the  surveyor  is  to  retrace  old  boundary  lines,  find  old 
corners,  and  relocate  them  when  lost.  In  performing 
this  duty,  he  exercises,  to  a  certain  extent,  judicial 
functions.  He  usually  takes  the  place  of  both  judge 
and  jury,  and  acting  as  arbiter  between  adjoining  pro- 
prietors, decides  both  the  law  and  the  facts  in  regard 
to  their  boundary  lines.  He  does  this  not  because  of 
any  right  or  authority  he  may  possess,  but  because  the 
interested  parties  voluntarily  submit  their  differences 
to  him  as  an  expert  in  such  matters,  preferring  to 
abide  by  his  decision  rather  than  go  to  law  about  it. 

In  making  resurveys  the  surveyor  is  called  upon  — 

1.  To  construe  descriptions  in  deeds; 

2.  To  find  the  location   of   corners  and  boundary 
lines ; 

3.  To  renew  corner  monuments  and  to  mark  anew 
boundary  lines. 

2.  In  construing  the  descriptions  the  following 
rules  have  been  laid  down  by  the  courts: 

RULE  1.  The  description  of  boundaries  in  a  deed  is 
to  be  taken  most  strongly  against  the  grantor. 

Marshall  v.  Niles,  8  Conn.  369. 
Ryan  v.  Wilson,  9  Mieh.  262. 

2.  Where  a  deed  contains  two  descriptions,  each 
complete  in  itself,  of  the  land  conveyed, —  one  of  the 
descriptions  including  all  the  land  included  in  the 
other,  and  more  besides, —  the  deed  passes  title  to  all 
the  land  contained  in  the  larger  tract. 

Lake  Erie  &  W.  R.  Co.  v.  Whitman  (111.  Sup.)  40  N.  E.  1014. 
19 


290  A    MANUAL    OF    LAND    SURVEYING. 

3.  In  ascertaining  boundaries  from  title  papers,  he 
who  has  the  oldest  title  is  entitled  to  take  his  courses 
and  distances,  go  where  they  may. 

Quillen   v.    Betts    (Del.    Super.    1897),   39   A.    595. 

4.  A  deed  must  be  construed  according  to  the  con- 
dition of  things  at  the  date  thereof. 

Crogan  v.  Burling  Mills,  124  Mass.  390. 

5.  Written  descriptions  of  property  are  to  be  inter- 
preted in  the  light  of  the  facts  known  to  and  in  the 
minds  of  the  parties  at  the  time. 

Wiley  v.    Sanders,    36   Mich.   60. 
McConnell  v.  Rathbun,  46  Mich.  305. 

6.  And  should  be  construed  with  reference  to  any 
plats,  facts,  and  monuments  on  the  ground  referred 
to  in  the  instrument. 

Anderson  v.   Baughman,  7   Mich.   77. 
Bowen  v.   Earl,  28  Mich.   538. 

7.  In    construing   a   deed,   the   court   will   consider 
the  whole  instrument,  and  when  the  calls  in  a   deed 
lead  to  conflicting  results,  that  construction  must  be 
adopted  which  is  most  consistent  with  the  intent  ap- 
parent on  its  face. 

Hitchler  v.  Boyles  (Tex.  Civ.  App.  1899),  51  S.  W.  648. 

8.  The  intention  of  the  parties  is  as  much  a  gov- 
erning rule  in  construing  a  deed  as  a  will. 

9.  In  construing  the  provisions  of  a  deed,  the  cir- 
cumstances of  the  parties  and  the  property  are  always 
elements  of  interpretation. 

Snowden  v.  Cavanaugh  (Pa.  Com.  PI.  1900),  10  Kulp,  1. 

10.  The  evident  intention  of  parties  to  a  deed  must 
be  given  effect,  the  same  as  in  other  writings. 

Hale   v.   Docking   (Kan.   App.   1897),   51    P.   798. 

11.  In  determining  the  true  construction  of  a  deed, 
the  habendum  clause  is  not  absolutely  controlling.  The 


RESURYEYS.  291 

real  question  is,  What  was  the  intention  of  the  grantor, 
to  be  gathered  from  the  entire  instrument,  and  not 
inconsistent  with  any  rule  of  law?  1  Rev.  St.  P. 
748,  §  2. 

Harriot  v.   Harriot,   49   N.    Y.    S.   447. 

12.  Ordinarily  the  intent  which   is  effective  in  a 
grant  is  the  intent  expressed  in  its  language,  and  is 
to  be  ascertained  by  giving  suitable  effect  to  all  the 
words  of  the  grant,  read  in  the  light  of  the  circum- 
stances attending  the  transaction,  the  situation  of  the 
parties,  and  the  state  of  the  country  and  of  the  estate 
granted. 

Proctor  v.  Maine  Cent.   R.   Co.,  52  A.   933,  96  Me.   458. 

13.  Where  the  description  of  the  boundaries  in  a 
deed    are    indefinite    or    uncertain,    the    construction 
given  by  the  parties,  and  manifested  by  their  acts  on 
the  ground,  is  deemed  the  true  one  unless  the  contrary 
is  clearly  shown. 

Reed  v.  Prop.  Locks  and  Canals,  8  How.   (U.   S.)  274. 

14.  Every  call  in  the  description  of  the  premises 
in  a  deed  must  be  answered  if  it  can  be  done,  and 
none  is  to  be  rejected  if  all  the  parts  can  stand  con- 
sistently together. 

Herrick  v.  Hopkins,  10  Shep..    (Me.)   217. 

15.  Where  the  boundaries  mentioned  are  inconsis- 
tent with  each  other,  those  are  to  be  retained  which 
best  subserve  the  prevailing  intention  manifested  on 
the  face  of  the  deed. 

Gates  v.   Lewis,  7   Vt.  511. 

16.  The  certain  description  must  prevail  over  the 
uncertain,  in  absence  of  controlling  circumstances. 

Richer  v.    Barry,  34  Me.   116. 

Tewksbury  v.   French,   44  Mich.   102. 

See  also  35  N.  H.  121,  and  11  Conn.  335. 


f      292  A    MANUAL    OF    LAND    SURVEYING. 

17.  The  rule  is  that  as  soon  as  there  is  an  adequate 
and    sufficient    definition,    with    convenient    certainty, 
of  what  is  intended  to  pass  by  the  particular  instru- 
ment,   a   subsequent   erroneous   addition   will   not   vi- 
tiate it. 

Airey  v.  Kunkle  (Com.  PL),  18  Pa.  Dist.  R.  620,  6  Pa.  Dist.  R.  1. 

18.  When  one  part  of  the  description  in  a  deed  is 
false  and  impossible,  but  by  rejecting  that  a  perfect 
description    remains,    such   false   and   impossible   part 
should  be  rejected  and  the  deed  held  good. 

Anderson  v.   Baughman,  7  Mich.  79. 
Johnson  v.   Scott,  11  Mich.  232. 

19.  A  deed  is  to  be  construed  so  as  to  make  it  ef- 
fectual rather  than  void.     (Ibid.) 

20.  General  words*  in  the  habendum  cannot  control 
or   govern    special    words    of   limitation   used    in    the 
grant  or  premises  of  a  deed. 

Hunter   v.    Patterson    (Mo.    1898),   44   S.   VV.   250. 

21.  Where  the  description  in  a  deed  calls  for  land 
"owned  and  occupied,"  the  actual  line  of  occupation 
is  a  material  call  to  be  considered  in  locating  the  lines 
Of  the  land  bounded  therein. 

Fahey  v.  Marsh,  40  Mich.  239. 
Cronin  v.  Gore,  38  Mich    386. 

22.  Where  land  is  described  as  running  a  certain 
distance  by  measure  to  a  known  line,  that  line  will 
control  the  measure  and  determine  the  extent  of  the 
grant. 

Flagg  v.  Thurston,  13  Pick.    (N.  Y.)    145. 

See  also  13  Wend.    (N.  Y.)   300,  and  7  Iredell.    (N.   C.)    169  and 
310. 

23.  Where    a  patent   calls   for  unmarked  lines   of 
surrounding   surveys,    the  position   of   which    can   be 
accurately  ascertained,  and  there  is  no  evidence  as  to 
how  the  survey   was   actually   made,   such   unmarked 


RESUKVEYS.  293 

lines  will  prevail  over  courses  and  distances,  in  case 
of  a  conflict. 

Maddox  v.   Turner   (Tex.),  15  S.  W.  237. 

24.  Not  so  if  the  line  is  obscure,  not  definitely  fixed, 
marked  or  known,   and  therefore  likely  to  be  looked 
upon  by  the  parties  as  less  certain  than  the  measure 
given. 

Howell  v.  Merrill,  30  Mich.  282. 

25.  In  the  case   of  Land  Co.  v.   Saunders  in   5th 
Otto  (U.  S.),  the  Supreme  Court  of  the  United  States 
held  the  west  line  of  Hart's  location  to  be  the  boundary 
of  a  grant.    It  was  in  a  mountainous  country  and  had 
never  been  surveyed  or  marked  —  although  capable  of 
being  marked  —  the  line  being  simply  marked  on  the 
plat  of  the  location.     This  -line  is  held  to  be  such  a 
monument  as  would  control  course  and  distance. 

26.  Where  land  is  conveyed  "  beginning  at "   and 
bounding  land  of  "  B,"   the  point  of  beginning  and 
boundary  is  the  true  line  of  B's  land,  and  -not  the  line 
of  occupation  as  shown  by  a  fence  set  up  and  main- 
tained by  B  before  and  after  the  conveyance,  with  the 
consent  of  the  owner  of  the  lot  conveyed,  under  the 
mistaken  belief  that  such  was  the  true  line. 

Cleveland  v.  Flagg.  4  Gushing  (Mass.)   76. 

27.  A  course  from  corner  to  corner  means  prima 
facie  a  right  line;  but  this  may  be  explained,  by  other 
matters  in  the  case,  to  be  a  crooked  or  curved  line, 
as  following  a  ditch,  or  hedge,  or  stream. 

Baker  v.  Talbott,  6  Mont.    (Ky.)   182. 

28.  "Northward"  or  "northerly"  means  due  north 
when  nothing  is  mentioned  to  show  the  deflection  of 
the  course  to  the  east  or  west. 

Jackson  v.  Reeves,  3  Caines,  N.   Y.   293. 
Brandt  v.  Ogden,  1  Johns.  N.  Y.  156. 


294  A   MANUAL    OF    LAND    SURVEYING. 

29.  The  use  of  the  term  "  about "   indicates  that 
exact  precision   is  not   intended;   but  where   nothing 
more  certain  can  be  found  to  control  the  course,  and 
distance,   the  grantee  is  limited  to  the  exact  course 
and  distance  given. 

Cutts  v.    King,    3   Greenl.    Me.    482. 

30.  Where  a  given  quantity  of  land  is  to  be  laid  off 
on  a  given  base,  it  must  be  included  in  four  lines,  so 
that  the  lines  proceeding  from  the  base   shall  be  at 
right  angles  with  it,   and  the  line  opposite  the  base 
shall  be  parallel  with  it,  unless  this  form  is  repugnant 
to  the  entry. 

Massie  v.  Watts,  6  Cranch.   (U.  S.)  148. 
Ker  v.  Watts,  6  Wheat.   (U.   S.)   550. 

31.  A  deed  described  land  conveyed  as  "  fractional 
N.  W.  corner  of  S.  W.  V4  of  section  6,  T.  13,  K.  6, 
containing   thirty-three    acres."      Held,   that   this    de- 
scription operated  to  convey  such  a  part  of  the  S.  W. 
1/4  section  6  as  was  included  between  lines  drawn  an 
equal  distance  from  the  point  of  intersection  of  the 
lines  bounding  the  quarter  section  at  its  N.  W.  cor- 
ner,   and    extended    a    sufficient    distance    to    include 
thirty-three  acres  between  parallel  lines,  and  that  the 
word  "  fractional "   should   either  be   treated   as   sur- 
plusage or   as   modifying   "  section,"    as   a    fractional 
corner  is  a  contradiction  in  terms. 

Swan  v.   New  England  Mortg.   Sec.   Co.    (Miss.  1898),  23   So.  627. 

32.  Seventy  acres   lying  and   being  in  the,  south- 
west corner  of  a  section,  is  a  good  description,  and 
the  land  will  be  in  a  square. 

Walsh  v.  Ringer,  2  Ham.   (Ohio)  327. 

33.  Where  lines  are  laid  down  on  a  map  or  plan, 
and   are    referred   to    in    a    conveyance    of   land,   the 
courses,  distances,  and  other  particulars  appearing  on 
such  plan  are  to  be  as  much  considered  the  true  de- 


RESURVEYS.  295 

scription  of  the  land  conveyed  as  they  would  if  ex- 
pressly recited  in  the  deed. 

Davis  v.   Rainsford,  17  Mass.   211. 

See  also  14  Mass.  149,  and  1st  Greenl.  Me.  219. 

34.  A  conveyance  by  metes  and  bounds  will  carry 
all  the  land  included  within  them,  although  it  be  more 
or  less  than  is  stated  in  the  deed. 

Butler  v.  Widger,  7  Cow.    (N.   Y.)    723. 
Bratton  v.  Clawson,  3   Strobh.   S.   C.  127. 
Gillman  v.  Riopelle,  18  Mich.  164. 

35.  A  definite   description  in  a  deed,  naming  the 
point  of  beginning,  the  monuments,  and  courses  and 
distances,  followed  by  a  statement  as  to  the  number  of 
acres  conveyed,  passes  only  the  quantity  of  land  in- 
cluded in  the  specified  boundaries,  though  that  is  less 
than  the  number  of  acres  stated. 

Silver   Creek  Cement   Corp.  v.   Union   Lime  &  Cement  Co.    (Ind. 
Sup.)  35  N.  E.  125. 

36.  A  grant  of  land  bounded  by  a  highway  takes 
to  the  center  of  the  highway.     If  it  be  designed  to 
exclude  the  highway,  it  must  be  so  stated  in  explicit 
terms. 

Champlin  v.   Pendleton,  13  Conn.  23. 
See  also  7  N.  H.  275;  6  Shep.  Me.  276. 
Purkiss  v.  Benson,  28  Mich.  538. 

37.  A  deed  of  land  lying  east  of  a  certain  street,  and 
explicitly  bounded  by  the  east  line  of  the  street,  con- 
veys no  title  to  the  soil  in  the  street. 

G.  R.  &  I.  R.  R.  Co.  v.  Mary  Heisel,  38  Mich.  62. 

38.  The  fact  that  a  patent  describes  the  land  as 
"  the  north-east  quarter  of  the  south-east  quarter  of 
section  8,"  instead  of  as  lot  4,  does  not  exclude  from 
the  grant  any  of  the  land  that  would  properly  be  in 
lot  4,  when  it  appears  by  the  government  plat  that 
it  was  intended  to  pass  lot  4,  which  was  marked  as 
containing  the  same  number  of  acres  as  were  granted 
by  the  patent. 


296  A   MANUAL    OF    LAND    SURVEYING. 

39.  The  fact  that  lot  4  is  in  two  quarter  sections, 
is  immaterial,  as  a  given  fractional  lot  may  be  crossed 
by  quarter  section  lines,  when  the   lines  of  the  plat 
show  that  it  was  not  intended  to  extend  the  intersect- 
ing line  of  the  adjoining  quarter  section  so  as  to  ex- 
clude  from   lot  4   the  land  in  controversy,   which  is 
necessary  to  make  up  the  number  of  acres  as  called 
for  in  the  plat  for  lot  4. 

Sheppard  v.   Wilmott   (Wis.)    47   N.   W.   1054. 

40.  The   presumption    that    a    conveyance   of   land 
passes  one  half  of  the  soil  of  the  adjoining  highway 
extends  to  streets  in  towns,  and  is  not  rebutted  by  the 
circumstance  that  the  grantor  is  the  municipal  author- 
ity entitled  to  part  of  the  soil  of  the  other  half  of  the 
street. 

Eng.   1898. 

41.  In  construing  the  description  in  a  deed  which 
bounds  the  land  conveyed  upon  a  street,  river,  or  other 
monument   having   width,    courts   incline   strongly   to 
such  an  interpretation  of  the  language  as  will  carry 
the  fee  of  the  land  to  the  center  line  of  such  mon- 
ument, rather  than  to  its  edge  only. 

Paine    v.    Consumers'    Forwarding   &    Storage   Co.    (C.    C.   A.)    7J 
F.  626. 

42.  The  mention  of  quantity  of  acres  after  a  defi 
nite  description  by  metes  and  bounds,  or  by  the  aliquot 
part  of  the   section,  is  a  matter  of  description   only, 
and  quantity  being  the  least  certain,  does  not  control. 

Amich  v.   Holman,  13   Strobh.   S.    C.   132. 
McClintock  v.   Rogers,  11   Ills.   279. 
Martin  v.  Carlin,  19  Wis.  454. 

43.  Where  boundaries  are  doubtful,  then  quantity 
often  becomes  a  controlling  consideration. 

Winans   v.    Cheney,   55     Cal.    567. 

44.  Where  the  other  terms  of  "the  description  con- 
tained in  a  deed  are  not  sufficiently  certain,  the  num- 


RESURYEYS.  297 

ber  of  acres  given  is  an  essential  part  of  the  descrip- 
tion, and  may  be  resorted  to  in  aid  of  tbe  defective 
part  of  such  description. 

Campbell   v.   Carruth    (Fla.)    13   So.    432. 

45.  Grants  by  government  are  to  be  construed  ac- 
cording to  the  common  law,  unless  it  has  done  some 
act  to  exclude  that  construction. 

Middleton  v.   Pritchard,  3  Scam.  111.  510. 

46.  In  construing   a   deed  describing  land  by  the 
government  survey  the  court  must  ascertain  the  cor- 
ners of  the  survey  as   actually  established,   and  not 
as  they  ought  to  have  been  established;  but  the  pre- 
sumption that  the  deed  was  intended  to  convey  accord- 
ing to  the  established  corners  may  be  rebutted  by  evi- 
dence that  the  parties  were  mistaken  as  to  the  loca- 
tion of  the  government  line,  and  intended  to  convey 
a  definite  tract. 

Squire  v.   Greer   (Wash.)    26  P.   222. 

47.  A  reference  in  a  description  to  the  government 
patent,  makes  the  patent  description  and  the  govern- 
ment survey  a  part  of  the  deed. 

Miller  v.   Topeka   Land   Co.    (Kan.)    24  P.   420. 

48.  Where  a  survey  is  referred  to  in  a   deed  for 
greater  certainty,  it  legally  forms  a  part  of  it  and  both 
should  be  construed  together. 

Heffleman  v.  Otsego  Water  Power  Co.  (Mich.)  43  N.  W.  1096. 

49.  Extrinsic  evidence  is  always  admissible  to  ex- 
plain the  calls  of  a  deed  for  the  purpose  of  applying 
them  to  the  subject-matter,  and  thus  to  give  effect  to 
the  deed. 

Thompson  v.  Southern  Cal.  M.  R.  Co.   (Cal.)   23  P.  130. 

50.  The  locative  calls  in  a  patent,  in  accordance 
with  which  the  survey  is  admitted  to  have  been  actu- 
ally made  on  the  ground,  control  more  general  calls  a"s 


298  A    MANUAL    OF    LAND    SURVEYING. 

to  the  county,  and  the  direction  and  distance  of  the 
land  from  certain  well-known  points. 

Minor  v.  Kirkland   (Tex.  Civ.  App.)   20  S.  W.   932. 

51.  An  exception  in  a  deed  which  reads,  "  Except 
the  dower  of  fifty  acres,  as  fully  described  in  the  deed 
given  the  C.  B.  Co.,"  is  not  void,  though  the  boun- 
daries  of   the   excepted  land   are  not   defined   in  any 
way,  as  reference  may  be  had  to  the  deed  to  the  C.  B. 
Co.  to  ascertain  them. 

McAffee  v.  Arline  (Ga.)   10  S.  E.  441. 

52.  A  deed  conveying  property  by  lot  numbers  is 
not   void    for   uncertainty,   though   the   recorded   plat 
shows  no  division   of  the  blocks   into  lots;   it  being 
shown   that   the   proprietors   had    always    treated   the 
blocks  as  divided  into  lots,  and  that  for  many  years 
the  property  had  been  assessed,  conveyed,  and  generally 
known  by  the  lot  numbers. 

Marvin   v.    Elliott    (Mo.)    12    S.   W.   899. 

53.  Where  in  a  platted  block  the  lots  are  marked  on 
the  plat  as  having  the  same  number  of  front  feet  each, 
except  one,  the  specific  dimensions  of  which  are  also 
marked,    and   a   survey   shows   that   the    whole   block 
contains  more  front  feet  than  are  marked  on  the  plat, 
the  excess   must  be  distributed  between  all  the  lots, 
and  not  given  to  that  lot  only  which  differed  in  its  di- 
mensions from  the  rest. 

Pereles   v.    Magoon    (Wis.)    46    N.   W.    1047. 

54.  A  deed  of  lots   according  to  a  plat  on  which 
the  lots  are  bounded  on  one  side  by  an  alley,  passes 
title  to  the   center  of   the  alley,  where  the  grantor's 
title  extends  thereto.     Following  Schneider  v.  Jacob, 
5  S.  W.  350. 

Jacob  v.  Woolfolk  (Ky.)  14  S.  W.  415. 

55.  A  grantee  of  a  lot  in  a  recorded  plat,  unless  the 
terms  of  his  deed  or  the  plat  exclude  that  construction, 


BESUEYEYS.  299 

takes  to  the  center  of  adjoining  public  ways,  subject 
to  the  public  easement;  and  the  fact  that  the  de- 
scription in  his  deed,  after  stating  the  number  of  his 
lot,  gives  its  dimensions,  exclusive  of  the  highways, 
does  not  effect  such  construction. 

Brown  v.  City  of  Baraboo   (Wis.  1898)   74  N.  W.  223. 

56.  A    deed    describing    the    granted   premises    as 
"  subdivision  of  lot  No.  4   of  division  No.   16,"  etc., 
followed  by  the  total  number   of  acres   contained  in 
lot  4,  and  then  excepting  laud  previously  sold,  is  not 
void  for  indefiniteness,  though  lot  4  was  never  sub- 
divided, as  it  evinces  a  clear  intent  to  convey  the  bal- 
ance of  whatever  land  the  grantor  owned  in  lot   4; 
and  the  deed  will   be  construed  as  though  the   word 
"  of  "  after  the  word  "  subdivision  "  had  been  omitted. 

Weeks  v.  Martin,  10  N.  Y.  S.  656. 

57.  A  deed  to  a  railroad  company  of  a  right  of  way 
"  along  the  line  as  surveyed  and  laid  out "  by  the  com- 
pany's engineer  is  not  void  for  uncertainty  where  it 
appears  that  when  the  deed  was  executed  the  line  of 
the   road   had   been    surveyed    and    distinctly   marked 
by  stakes  stuck  in  the  ground,  and  that  subsequently 
the  road  was  constructed  following  the  exact  line  of 
the  survey. 

Thompson  v.  Southern  CaJ.  M.  R.  Co.   (Cal.)  23  P.  130. 

58.  A  deed  conveying  the  land  south  of  a  "  railway 
cut"  conveys  only  the  land  south  of  the  upper  and 
outer  edge  of  the  cut. 

Newton  v.  Louisville  &  N.  R.  Co.   (Ala.)  19  So.  19. 

59.  When  land  is  bounded  on  a  private  way,  it  ex- 
tends only  to  the  side  line  of  the  way. 

Winslow  v.    Reed,   35   A.    1017,   89   Me.    67. 

60.  In  a  deed  of  land  by  metes  and  bounds,  an  ex- 
ception, of  "  lot  6,  block  36,  heretofore  conveyed  to  B," 
excepts  a  lot   so   numbered  on   a   plat  •  made  by  the 


300  A    MANUAL    OF    LAND    SURVEYING. 

grantor  and  grantee,  but  not  then  recorded,  there  be- 
ing no  other  lot  6  block  36,  within  the  land  granted. 
The  recital  of  a  conveyance  to  B.  may  be  rejected  as  a 
falsa  demonstratio. 

Ambs  v.  Chicago,  St.  P.,  M.  &  O.  R'y    Co.   (Minn.)  46  N.  W.  321. 

61.  Where  a  deed  conveys  lots  according  to  a  cer- 
tain plat,  the  fact  that  the  plat  is  invalid  does  not 
affect  the  deed. 

Young  v.  Cosgrove   (Iowa)   49  N.  W.  1040. 

62.  Though   a   plat  be   incomplete   as  respects  the 
location  of  monuments,  or  in  respect  to  measurements 
and  distances,  yet  where  land  so   surveyed  has  been 
conveyed  by  reference  thereto,  and  the  location  of  the 
lots  so  conveyed  and  designated  is  well  known  by  all 
parties  interested,  and  susceptible  of  identification  ac- 
cording to  the  actual  survey  on  the  ground,  the   de- 
scription is  sufficient  to  pass  the  title. 

Bohrer  v.  Lange   (Minn.)   46  N.  W.   358; 

63.  The  description  in  a  deed  was :  "  Beginning  at 
*  ;   running  thence   northeasterly,   along  Grove 

street,  25  feet;  and  thence  northwesterly,  and  parallel 
with  Woodruff  avenue,  108  feet  9  inches,  to  lot  No.  80, 
on  said  map;  thence  southwesterly,  along  lot  No.  80, 
25  feet;  and  thence  southeasterly,  and  parallel  with 
Woodruff  avenue,  108  feet  9  in.,  to  the  westerly  side 
of  Grove  street,  the  point  or  place  of  beginning." 
Lines  drawn  from  Grove  street,  108  feet  9  inches,  par- 
allel to  Woodruff  avenue,  would  not  reach  lot  80 
by  5  inches.  Held,  that  there  was  a  mistake  in  de- 
scribing the  length  of  the  lines  parallel  to  Woodruff 
avenue,  and  that  it  was  intended  that  they  should  ex- 
tend 109  feet  2  inches,  and  not  that  they  should  run 
in  such  a  direction  that  they  would  reach  lot  80  at  the 
distance  of  108  feet  9  inches  from  Grove  street. 

Casey  v.  Dunn,  8  N.  Y.  S.  305. 


KESURVEYS.  301 

64.  It  being  stated  with  certainty  in  the  deed  that 
such  lines  were  parallel  to  Woodruff  avenue,  it  is  im- 
material, in  construing  the  description,  that  the  cor- 
responding  lines   in  the  conveyances   of  neighboring 
property  were  at  right  angles  to  Grove  street,  instead 
of  being  parallel  to  Woodruff  avenue. 

Casey  v.  Dunn,  8  N.  Y.  S.  305. 

65.  The   description  in  a  deed  was  certain   as  to 
the  northern  and  western  boundaries.     The  course  of 
the  eastern  boundary  was  south  for  a  distance  of  8 
rods.     The  southern  boundary  was  "  then  west,  in  a 
line  parallel  to,  and  eight  rods  south  of,"  the  north- 
ern boundary,  "  one  hundred  and  sixty-two  feet,  to " 
the  western  boundary.     By  reference  to  another  deed, 
it  was  made  certain  that  the  north  6  rods  of  the  eastern 
boundary  was  a  straight  wall.    The  course  of  the  other 
2  rods  was  uncertain.     Extending  the  line  of  the  wall 
2  rods  south,  and  from  the  end  of  this  line  drawing 
a  line  parallel  to  the  northern  boundary,  to  the  west- 
ern boundary,  a  southern  boundary  165  feet  in  length 
would  be  obtained.    Held,  that  from  the  southern  end 
of  the  wall  the  eastern  line  should  be  deflected  towards 
the  west  at  such  an  angle  that  at  the  distance  of  2 
rods  it  would  intersect  a  line  parallel  with  the  north- 
ern boundary  at  the  distance  of   162  feet  from   the 
western  boundary. 

Ladies'  Seamen's  Friend  Soc.  v.  Halstead   (Conn.)   19  A.  658. 

66.  A  city,  by  its  president  and  trustees,  conveyed 
to  defendants'  grantor  "that  lot  of  land  containing 
60  acres,  lying  in  block  No.  1111,  according  to  the  offi- 
cial map  of  said  city  made  by  *  *  *  A.  D.  1856." 
The  deed  referred  to  a  resolution  of  the  trustees,  un- 
der which  the  lands  were  sold,  which  provided  that 
all  surveys  should  be  made  by  the  purchaser.  At  the 
time  of  the  deed  there  had  been  no  survey  or  subdivi- 


302  A    MANUAL    OF    LAND    SURVEYING. 

sion  of  the  block.     Held,  that  the  deed  conveyed  an 
undivided  60  acres  of  the  block. 

Cullen  v.    Sprigg    (Cal.)    23   P.   222. 

67.  Where  a  description  by 'metes   and  bounds  is 
supplemented  by  a  reference  to  a  particular  subdivi- 
sion of  land  to  indicate  the  tract  intended  to  be  con- 
veyed, the  former  will  not  necessarily  be  controlling, 
when  it  would  leave  a  strip  13  feet  front  by  100  deep 
in   the  grantor,   which   clearly   appears   to   have   been 
intended  to  be  conveyed  by  the  latter  description. 

Cannon  v.  Emmons  (Minn.)   46  N.  W.  356. 

68.  Ordinarily,  calls  for  natural  or  artificial  mon- 
uments will  control  courses  and  distances;  but  a  call 
for  course  and  distance  will  not  be  subordinated  to  a 
call  for  an  unmarked  line  in  a  prairie,  which  cannot 
itself  be  ascertained  except  by  running  the  boundaries 
of  another  survey  according  to  course  and  distance. 

Johnson  v.  Archibald   (Tex.)   14  S.  W.  266. 

69.  A  complaint  was  filed  to  quiet  title  to  150  acres 
of  land  lying  on  the  south  side  of  a  fractional  section. 
A  surveyor  was  ordered  to  survey  that  quantity,  to  be 
taken  the,  full  length  of  the  section  from  the  east  side 
thereof  to  a   river  as  the  western  boundary,  and  ex- 
tending far  enough  north  to  include  150  acres.     The 
surveyor  executed  the  order,   and  reported   a   survey, 
which  was  accepted,  and  the  court  entered  judgment, 
wherein  the  land  was  doubly  described  by  inconsistent 
descriptions.     The  first  described   it   as   in  the  order 
of  survey,  and  the   second  by  metes  and  bounds,  by 
which,  after  beginning  at  the  southeast  corner  of  the 
section,   and   following    the    south   line   to   the   river, 
it  ran  up  the  river,  with  the  meanders  thereof,  to  a 
stake  placed  by  said  surveyor  19  1-3  chains  north  of 
the  south  line  of  the  section ;  thence  running  westerly, 
parallel  with  the  south  line,  53.04  chains,  to  a  stake 


EESURVEYS.  303 

in  the  east  line  of  the  section;  and  thence  southerly 
with  said  line  9 1-3  chains,  to  the  beginning.  The 
stakes  were  gone,  but  were  shown  to  have  been  placed 
at  points  19  1-3  chains  from  the  south  line,  thereby 
including  150  acres.  Held,  that  the  first  description 
should  govern. 

Caspar  v.  Jamison  (Ind.)  21  N.  E.  743. 

70.  Under  a  deed  of  land  bounded  by  a  street,  ac- 
cording to  a  map  referred  to,  the  line  of  the  street 
as  actually  surveyed  is  the  boundary  of  the  land  con- 
veyed. 

Andreu  v.  Watkins   (Fla.)  7  So.  876. 

71.  A  deed  described  the  land  conveyed  as  "  com- 
mencing on  the  S.  road  at  the  north-east  corner  of  the 
land  owned  by  S. ;   running  south,  to   the  south-east 
corner  of  said  S.'s  land,  two  acres;  from  thence,  east- 
erly and  parallel  with  said  S.  road,  two  acres;  thence 
running  northerly  two  acres,  until  it  strikes  said  road ; 
and  thence  westerly,  along  said  road,  two  acres,  to  the 
beginning;    containing    four    acres    of   land,    neither, 
more  nor  less."    Held,  that  as  the  description  by  quan- 
tity so  clearly  shows  the  intention  to  limit  the  grant 
to  four  acres  in  rectangular  form,  and  as  the  length 
of  the  west  line  is  given,  the  intention  must  control 
distances. 

Rioux  v.  Cormier  (Wis.)   44  N.  W.  654. 

72.  A  similar  construction  is  to  be  given  the  United 
States    statute   providing   for   the   survey    in    certain 
cases  of  tracts  of  land  two  acres  in  width  and  running 
back  a  depth  of  forty  acres.     E.  S.  2407. 

73.  A  city  condemned  a  strip  of  land  for  railroad 
and  sewer  purposes,  and,  after  constructing  a  road-bed 
along  this,  it  conveyed  to  a  railroad  company  "  its  title 
to  the  road  bed,  bridges,  and  right  of  way  "  along  the 
entire  route,  and  "  all  the  land  belonging  to  the  city," 


304  A    MANUAL    OF    LAND    SURVEYING. 

between  certain  streets,  "  for  depot  purposes."  The 
company  had  formerly  occupied  a  right  of  way  for  a 
double  track  on  other  streets,  and  the  city,  in  consid- 
eration of  the  change  of  the  railway  to  the  street  form- 
ing the  line  of  the  road  in  the  conveyance,  agreed  to 
furnish  the  company  a  road-bed.  Held,  that  outside 
of  the  part  conveyed  for  depot  purposes  nothing  but 
the  road-bed  was  conveyed. 

Long  v.  Louisville  &  N.   R.  Co.    (Ky.)   13  S.  W.  3. 

74.  The  deed  of  a  city  lot,  and  plat  with  reference 
to   which  it  was   made,  called  for  the'  south  line  of 
Cherry  street  as  the  northern  boundary  of  the  lot.     The 
line    referred    to    had   been    established    by    the    City 
Surveyor  37  years  before  and  ever  since  acquiesced  in. 
The  other  lots  in  the  block  had  been  bought,  fenced 
and  built  upon  on  the  assumption  that  this  survey  was 
correct.     A  more   recent  survey  tended  to   show  that 
the  line 'was  three  feet  too  far  north.     Held,  that  the 
presumption  of  correctness  was  with  the  older  survey, 
and  as  the  lot  owner  had  got  all  he  bargained  for,  and 
the  later  survey  would "  cause  the  lines  of  the  other 
lots  to  cut  into  the  buildings,  the  older  survey  must 
prevail. 

Wilmarth   v.   Woodcock,   Mich.    33   N.   W.    Rep.    401. 

75.  A  description  of  a  deed  reads :  "  The  east  ^2  of 
the  east  y%  of  the  northwest  i/4,  and  the  east  V%  of 
the  east  V2  of  the  southwest  frac.  y±"  etc.,  contain- 
ing 50  acres  of  land;  being  the  east  half  of  100  acres 
conveyed  by  A.  and  B.  to  E.     The  south  line  of  the 
tract   is   irregular   on    a   lake,   and  a  line  north   and 
south  through  the  center  of  the  tract  would  give  one 
parcel   nine   acres   more   land  than  the    other.     Held, 
that  the  language  is  apt  and  proper  to  divide  the  tract 
by  a  north  and  south  line  which  would  give  to  each 
50  acres,  or  one-half  of  the  whole. 


RESURVEYS.  305 

76.  A  description  of  the  half  of  the  parcel  of  land, 
according   to   the   United   States    survey,   would   have 
excluded  the  idea  of  equal  quantities  and  fixed  the  di- 
viding line  in  accordance  with  the  Act  of  Congress. 
If  any  other  line  had  been  agreed  upon  between  the 
owners  as  the  boundary  line,  it  would  govern  the  case. 

Jones  v.   Pashby,  Mich.  29  X.  W.  Rep.  376. 
Dart  v.    Barbour,  32   Mich.   276. 
Heyer  v.  Lee,  40  Mich.  353. 

77.  A  description  in  a  deed,  if  otherwise  good,  is 
not  vitiated  by  the  omission  of  the  word  "  rods "  to 
avoid  tautology,  when  the  meaning  is  plain. 

Taber  v.  Shattuck,  55   Mich.   370. 

78.  In  a  deed  of  the  "east  half"  of  a  fractional 
quarter  cection,  the  words   "east  half"   refer  to  the 
government  subdivision  of  a  quarter  section,  and  not 
to  a  subdivision  of  the  quarter  section  by  a  line  divid- 
ing it  into  two  equal  parts. 

Turner  v.  Union  Pac.  Ry.  Co.   (Mo.  Sup.)  20  S.  W.  673. 

79.  A  deed  of  the  "  west  half  "  of  a  fractional  lot 
containing  less  than  a  legal  subdivision  of  40  acres 
conveys  half  of  the  area. 

Owen  v.  Henderson  (Wash.)  47  P.  215. 

80.  Held,   that,   in   order   to   determine   the   south 
boundary  of  the  north  31  acres  of  a  lot  bounded  on  the 
east    by    a    lake    conveyed    to    defendant,    the    land 
should  be  measured  to  the  meandered  line  of  the  lake, 
and  not  to  the  water's  edge;   such  measurement  car- 
rying out  the  parties'  expressed  intention  of  dividing 
the  property  equally. 

Peck  v.   Webb,  88  N.   W.   888,  Mich.   1902. 

1.  Adverse  Possession. — When  the  boundary  line 
between  the  lands  owned  by  adjoining  land-owners  is 
unknown,  they  may  by  parol  fix  a  line  between  each 
party,  each  party  mutually  agreeing  thereto  and  act- 
ing thereon,  which  is  binding  between  them;  but  if 
20 


306  A  MANUAL  OF  LAND  SURVEYING. 

tlje  line  is  known,  then  the  transfer  of  any  portion 
of  the  land  on  one  side  of  the  line  from  the  one  to 
the  other  must  be  in  writing,  to  be  valid. 

Jenkins  v.   Trager,  40   F.   726. 

2.  The  adverse  possession  of  land  by  a  grantor  can- 
not avail  his  grantee,   beyond  the   boundary  line   de- 
scribed in  the  deed. 

Jenkins  v.   Trager,   40   F.   726. 

3.  Possession  as  owner  is  an  essential  condition  by 
which  the  ownership  of  immovables  can  be  acquired 
without  title,  or  possession  in  good  faith. 

Stille  v.    Schull    (La.)    6   So.    634. 

4.  Continuous  possession  of  land  for  more  than  30 
years  under  claim  of  ownership,  though  without  color 
of  title,  constitutes  title  in  fee. 

Bowen  v.   Swander   (Ind.)   22  N.   E.   725. 

5.  One  cannot  acquire  title  to  land  by  adverse  pos- 
session where  he  claims  title  under  a  deed  which  in 
fact  does  not  include  such  land  in  its  description. 

Casey  v.  Dunn,  8  N.  Y.  S.  305. 

6.  Where    title   is    claimed   by    adverse   possession, 
if  the  possession  is  by  actual  occupation  of  the  pos- 
sessor under  claim  of  title,  it  is  visible,  open,  notorious, 
distinct,  and  will  be  presumed  to  be  hostile. 

Green  v.   Anglemire    (Mich.)    43   N.   W.   772. 

7.  Where  the  line  between  adjoining  owners  is  in 
doubt,  but  they  only  claim  ownership  to  the  true  line, 
wherever  that  may  be,  no  title  by  adverse  possession 
can  arise  in  either,  as  against  the  other. 

Krider  v.   Milner   (Mo.)   12   S.  W.   461. 

3.  In  construing  deeds  conveying  title  to  lands 
bordering  on  waters,  it  will  be  necessary  for  the  sur- 
veyor to  inquire  into  the  local  laws  of  the  State  in 
which  the  premises  lie,  as  different  States  by  their  laws 
and  courts  give  different  constructions  to  the  word 


LANDS    BORDERING    ON    WATERS.  307 

"  navigable "   as   applied   to   streams   and  the   smaller 
lakes.     The  statute  of  the  iTnited  States  provides  that 

"SEC.  440.  All  navigable  rivers,  within  the  territory  occupied  by 
the  public  lands,  shall  remain  and  be  deemed  public  highways;  and, 
jn  all  cases  where  the  opposite  banks  of  any  streams  not  navigable 
belong  to  different  persons,  the  stream  and  the  bed  thereof  shall  be- 
come common  to  both/'  (1  Stat.  468;  2  id.  235;  R.  S.  2476.) 

It  is  a  universal  rule  that  grants  of  land  border- 
ing on  navigable  streams  take  only  to  high-water  mark, 
while  grants  on  non -navigable  streams  take  to  the  cen- 
ter of  the  stream,  or  the  filum  aquas,  as  it  is  termed. 

Now,  whether  the  proprietor  in  any  given  case 
owns  the  land  under  water  to  the  center  of  the  stream, 
or  only  takes  to  high-water  mark,  depends  on  the  local 
construction  given  to  this  word  navigable. 

Under  the  Common  Law,  a  navigable  stream  is 
one  in  which  the  tide  ebbs  and  flows.  Some  excep- 
tions to  the  rule  are  made  in  England. 

Under  the  Civil  Law,  a  navigable  stream  is  one 
capable  of  being  used  as  a  highway  of  commerce.  In 
the  case  of  the  Railroad  Co.  v.  Schurmier  (7  Wallace, 
272),  the  Supreme  Court  of  the  United  States  says 
that  "  the  words  navigable  and  non-navigable  were 
applied  by  Congress  without  respect  to  the  ebb  and 
flow  of  the  tide,"  and  in  the  case  of  Bowman  and 
Burnley  v.  Wathieu  and  others,  (2d  McLean,  276), 
they  say  that  "  the  common  law  doctrine  as  to  the 
navigableness  of  streams  can  have  no  application  in 
this  country,  and  the  fact  of  navigableness  does  in 
no  respect  depend  on  the  ebb  and  flow  of  the  tide." 

The  courts  of  Pennsylvania,  North  Carolina, 
South  Carolina  and  Alabama  hold  the  same  view.  On 
the  contrary,  in  Maine,  New  Hampshire,  Massachu- 
setts, Connecticut,  New  York,  Maryland,  Virginia, 
Ohio,  Illinois,  Indiana,  and  Michigan,  the  common 
law  doctrine  is  held  to  prevail.  (See  Angell  on  Tide. 
Waters,  pp.  77  and  78.) 


A  MANUAL  OF  LAND  SURVEYING. 

Hence,  in  applying  the  principles  laid  down  by 
the  courts  in  the  following  decisions,  the  surveyor  will 
bear  in  mind  the  locality  in  which  they  are  to  be 
applied. 

1.  The  grants  of  the  government  for  land  bounded 
on  streams  and  other  waters  without  any  reservation 
or  restriction  of  terms  are  to  be  construed  as  to  their 
effect  according  to  the  law  of  the  state  in  which  the 
lands  lie. 

U.   S.   Sup.  Ct.,  Hardin  v.  Jordan,  29  N.  \y.  Rep.  813 

2.  Proprietors  of  lands  bordering  on  navigable  riv- 
ers, under  titles  derived  from  the  United  States,  hold 
only  to  the  stream,  as  the  express  provision  is,  that 
all  such  rivers  shall  be  deemed  to  be  and  remain  public, 
highways. 

R.  R.  Co.  v.  Sch,urmeir,  7  Wallace  (U.  S.)  272. 

3.  Where  a  sea  or  bay  is  named  as  a  boundary,  the 
line  of  ordinary  high-water  mark  is  always  the  line, 
where  the  common  law  prevails. 

U.  S.  v.  Pacheco,  2  Wallace  (U.  S.)  587. 

4.  A  boundary  on  a  stream  or  l>y  or  to  a  stream 
includes  flats  at  least  to  low-water  mark,  and  in  many 
cases  to  the  middle  thread  of  the  river. 

Thomas  v.   Hatch,   3   Sumner    (U.    S.)    170. 

5.  A  boundary  on  the  bank  of  a  river  referring  to 
fixed  monuments  on  the  bank,  limits  the  grant  to  the. 
bank  and  excludes  the  flats.     (Ibid.) 

See  also  Hopkins  v.  Kent,  9  Ohio,  13. 

6.  The  words  "along  the  bank"   are  strong  and 
definite  enough  to  exclude  the  idea  that  any  part  of 
the  river  or  its  bed  was  granted  in  the  navigable  or 
innavigable  parts  of  the  river. 

Howard  v.  Ingersoll,  13  How.  (U.  S.)  341,  416 

7.  A  deed  describing  the  land  by  a  boundary  run- 
ning to  a  stream,  and  thence  along  its  bank,  and  re- 
serving the  right  to  use  the  river  front  a  specified  time, 


LANDS    BORDERING    ON    WATERS.  309 

conveys  the  land  to  the  water's  edge  and  covers  the 
riparian  rights  to  the  middle  of  the  stream. 

Cole  v.   Wells,   49  Mich.   450. 

8.  Congress,    in    making    a    distinction    between 
streams  navigable  and  those  not  so,  in  the  acts  relating 
to  the  sale  of  the  public  lands  bordering  thereon,  in- 
tended to  provide  that  the  common  law  rules  of  ripa- 
rian ownership  should  apply  to  the  lands  bordering  on 
the  latter,  but  that  the  title  to  lands  bordering  on  the 
former  should  stop  at  the  stream. 

R.   R.   Co.  v.   Schurmeir,  7  Wall.    (U.  S.)   272. 

9.  In  streams  which  are  not  navigable,   adjacent 
proprietors  own  to  the  center  of  the  stream  measured 
from  low-water  mark. 

Clark  v.   Caupau,  19  Mich.  325. 

Moore  v.    Sanborn,   2   Mich.   519. 

Lorman  v.  Benson,  8  Mich.  18. 

Bay  City  Gas  Light  Co.  v.  Ind.  Wks.  8  Mich.  182. 

Lamb  v.  Ricketts,  11  Ohio  311. 

10.  The  same  principle  is  applied  to  Lake  Muske- 
gon,  in  Michigan    (Rice  v.  Ruddeman,  10  Mich.  125), 
but  not  applied  to  a  similar  lake  in  Wisconsin,  where 
the  court  says    (Deidrich   v.  K  W.  U.   Ry.   Co.,  42 
Wis.  271)  :  "  Riparian  owners  upon  a  natural  lake  or 
pond  take  only  to  the  shore." 

11.  In  the  case  of  the  State  of  Indiana  v.   Milk, 
Circuit  Court  of  the  United  States,  April  term,  1882, 
llth  Bissell,  page  197,  the  court  rejects  the  theory  of 
riparian  ownership   in  the  lake,  and  after  presenting 
its  reasons  at  some  length,  concludes  with  the  follow- 
ing :     "  That  while  a  general  grant  of  land  on  a  riv«r 
or  stream  non-navigable  extends  the  line  of  the  grantee 
to  the  middle  or  thread  of  the  current,  a  grant  on  a 
natural  pond  or  lake  extends  only  to  the  water's  edge." 

12.  Islands  in  rivers  fall  under  the  same  rule  as 
to  ownership  as  the   soil  under  water  does.     If   not 


310  A    MANUAL    OF    LAND    SURVEYING. 

otherwise  lawfully  appropriated,  they  belong  to .  the 
proprietors  on  either  side  of  the  stream,  according  to 
the  original  dividing  line  or  filum  aquce  as  it  would 
run  if  the  islands  were  under  water.  The  filum  aquce 
is  midway  between  the  lines  of  ordinary  low-water 
mark,  without  regard  to  the  channel  or  depth  oi 
water.  When  the  island  is  appropriated,  the  boundary 
is  then  midway  between  it  and  the  mainland. 

McCullough  v.   Wall,   4   Rich.    (S.   C.)    68. 
Kimball  v.   Schaff,  40  N.  H.  190. 

13.  The  grant  includes  any  land  between  the  mean- 
der line  and  the  water,  in  an  unnavigable  stream. 

The    same   principle    applies   to    unnavigable    lakes 

Forsyth  v.   Smale,  7  Biss.    (U.   S.)   201. 

14.  The  owners  of  land  bordering  on  the  shore  of  a 
meandered  non-navigable  or  dried-up  lake,  own  the  bed 
of  the  lake  in  severalty,  and  their  title  extends  to  the 
center ;  the  boundary  lines  of  each  abutting  tract  being 
fixed  by   extending,  from   the  meander  line  on   each 
side  of  the  tract,  lines  converging  to  a  point  in  the 
center  of  the  lake. 

Shell  v.  Matteson   (Minn.  1900)  83  N.  W.  491. 

15.  Where  an  island  springs  up  in  the  midst  of  a 
stream,  it  is  an  accretion  to  the  soil  in  the  bed  of  the 
river,  and  not  to  the  land  of  the  riparian  owner. 

East  Omaha  Land  Co.  v.  Hansen,  90  N.  W.  705   (Iowa,  1902). 

16.  Where,    after    submergence,    the    water    disap- 
pears from  the  land,  either  by  gradual  retirement  or 
elevation  of  the  land  by  natural  or  artificial  means, 
and    its    identity    can    be    established    by    reasonable 
marks,   or   by   situation    or  boundary   lines,   the   pro- 
prietorship returns  to  the  original  owner. 

Hughes  v.  Birney's  Heirs,  32  So.  30   (La.  1902). 

17.  High-water  mark  in   the  Mississippi  River  is 
to  be  determined  from  the  river  bed,  and  that  only  is 


LANDS    BORDERING    OX    WATERS.  311 

river  bed   which  the  river   occupies   long   enough   to 
wrest  it  from  vegetation. 

Houghton  v.   Railway  Co.,  47  Iowa    370. 

18.  A  bank  is  the  continuous  margin  where  vege- 
tation ceases.     The  shore  is  the  sandy  space  between 
it  and  low- water  mark. 

McCull«ugh  v.  Wainwright,  14  Penn.   St.   59. 

19.  Where  a  levee  was  shown  to  have  been  judi- 
ciously located  by  a   competent  engineer  and  agents 
of  the  State  acting  under  authority  conferred  by  the 
State  Legislature,  it  was  held  that  such  levee  became 
the  boundary  line  of  high  water,  and  that  no  private 
ownership   ceuld   be  acquired   to  land  lying  between 
that  and  the  bed  of  the  stream. 

Musser  v.   Hershey,   42   Iowa     356. 

20.  Grant  of  a  city  lot  bounded  on  a  river,  takes 
to  the  center  of  the  stream. 

Watson  v.   Peters,  26  Mich.  508. 

21.  Riparian  rights,   unless   expressly   limited,   ex- 
tend to  the  middle  of  the  navigable  channel,  and  cover 
any  shallows  or  middle  ground  not  shown  in  the  gov- 
ernment surveys,  but  lying  between  such  shallows  and 
the  shore,  and  it  makes  no  difference  that  the  deed 
conveying   the  premises   to    which   the   rights   attach 
describes   them   according   to   a   city  plat  instead  of 
the  government  entry. 

Fletcher  v.   Thunder  Bay   Boom  Co.,  51  Mich.  277. 

22.  But  if  the  plat  plainly  indicates  the  proprie- 
tor's intent  to  reserve  the  space  between  the  shore  and 
the  thread  or  main  channel,  the  case  would  be  dif- 
ferent. 

Watson  v.   Peters,  26  Mich.   508. 

23.  Riparian  rights  extend  laterally  into  the  stream. 
Rocks  and  shoals  along  the  margin  of  navigable  riv- 
ers above  tide-water  belong  to  the  riparian  owner. 

Moore  v.  Willamette  T.  &  L.  Co.,  7  Oregon  R.  355. 


312  A    MANUAL    OF    LAND    SURVEYING. 

24.  When  a  navigable  stream  is  meandered  in  mak- 
ing the  public  surveys,  and  the  United  States   has" 
granted  to  the  meander  line,  the  grantee  takes  to  the 
river.     The  stream,  and  not  the  meander  line,  is  the 
true  boundary  of  the  riparian  owner. 

Minto  v.  Delaney,  id.,  337. 

25.  Lands  patented  by  the  United  States  on  a  tide- 
water   stream   extend   to   the   meandered   line    of   thev 
stream,  which  is  the  line  of  ordinary  high  water. 

Parker  v.   Taylor,  id.,   435. 

26.  A  boundary  by  the  shore  of  a  mill  pond  takes 
to  low  water  mark. 

Stevens  v.   King,   76   Maine  197. 

27.  The   fact  that   a   deed    described   the  property 
conveyed  as  commencing  at  a  known  monument  on  the 
shore    of   a    pond,    and   running   thence   "  along    said 
pond,"  does  not  show  an  intention  to  convey  only  to 
the  shore. 

A  deed  of  land  bordering  on  a  small  non-navigable 
lake  or  pond  is  presumed  to  convey  title  to  the  center 
of  the  lake  or  pond,  unless  the  contrary  appears. 

Gouverneur  v.  National  Ice  Co.    (N.  Y.  App.)   31  N.   E.  865. 

28.  N.  conveyed  a  lot  according  to  a  certain  piat. 
The  plat  represented  the  lot  as  bounded  north  by  a 
street;  south  by  a  stream;  on  the  east  and  west  by 
lines  running  from  the  street  to  the  stream,  with  fig- 
ures purporting  to  give  the  length  of  these  -lines.     In 
fact,  the  distance  to  the  stream  was  greater  than  in- 
dicated by  these  figures.     Held,  that  the  conveyance 
of  the  lot  according  to  the  plat  included  all  the  land 
between  the  street  and  the  stream. 

Nicolin  v.  Schneiderham,  Minn.  33,  N.  W.  Rep.  33. 

29.  In  Turner  v.  Holland,  the   Supreme   Court  of 
Michigan    gives    riparian    rights    to    owners    of    lots 


LANDS    BORDERING    ON    WATERS.  313 

bounded  by  a  bayou  of   Saginaw  river,  described  by 
plat  similar  to  the  above. 

33  N.  W.  Rep.  283. 

30.  In  a  navigable  stream,  as  the  DesMoines  river 
in  Iowa,  high  water  mark  is  the  boundary  line.    When, 
by  action  of  the  water,  the  river  bed  changes,  high 
water  mark  changes  and  ownership  of  adjoining  land 
changes  with  it.     The  location  of  meander  lines  does 
not  affect  the  question.     Meander  lines  are  not  boun- 
dary lines. 

'Steele  v.  Sanchez,  33  X.  W.  Rep.  367. 
Krant  v.  Crawford,  10  Iowa  549. 
Lockwood  v.   R.   R.   Co.,  37  Conn.  387. 

31.  A  boundary  stated  in  a   deed   as  a  line  forty 
feet  above  the  border  of  a  river  at  high  water  mark,  is 
not  ambiguous,  and  if  disputed  is  to  be  fixed  like  any 
other  facts,  by  testimony  and  an  examination  of  the 
ground. 

Bresler  v.  Pitts,  59  Mich.  348. 

32.  A  patent  for  a  fractional  quarter  section,  which 
is  bounded  by  a  meandered  stream,  passes  title  to  all 
land  within  the  lines  of  said  quarter  section  between 
the  meandered  line  and  the  water's  edge. 

Sphung  v.  Moore,   (Ind.)   22  N.  E.  319. 

33.  The  owner  of  land  on  the  margin  of  a  naviga- 
ble  stream,   holding  under  a  grant  from  the  United 
States,  does  not  take  to  the  middle  of  the  stream,  but 
to  high  water  mark,  which  is  determined  by  the  change 
in  the  vegetation  and  the  character  of  the  soil,  and  the 
beds  of  all  navigable  streams,  though  the  tide  does  not 
ebb  and  flow  in  them,  belong  to  the  state. 

St.   Louis,   I.   M.   &   S.    Ry.   Co.  v.   Ramsey   (Ark.)    13   S.   W.   931. 

34.  The  owner  of  land  on  a  bay  conveyed  an  acre 
at  the  end  of  the  tract  nearest  the  bay,  described  as 
follows :    "  Beginning     *     *     *     by    the    beach,    run- 


314  A    MANUAL    OF    LAND    SURVEYING. 

ning  *  *  *  along  the  beach  to,"  etc.  In  the 
general  description  of  the  tract  it  was  bounded  "  east- 
erly by  the  said  beaph."  The  grantee  was  given  the 
privilege  of  a  road  from  the  middle  of  the  front  of 
the  lot  to  the  bay,  and  also  half  the  drift  coming  on 
shore  in  front  of  the  lot,  and  all  the  other  privileges 
of  the  beach  were  reserved  by  the  grantor,  who  bound 
himself  not  to  build  any  house  in  front  of  the  lot. 
The  courses  and  distances  would  not  carry  the  boun- 
dary to  high-water  mark.  Held,  that  the  beach  did 
not  pass  by  the  deed. 

Benson  v.  Townsend,  7  N.  Y.   S.  162. 

35.  Part  of  a  quarter  section  of  land  conveyed  was 
covered  by  a  lake.     The  deed  described  the  part  con- 
veyed as  140  acres  in  the  east  part  of  said  quarter  sec- 
tion.    Held,  that  the  deed  was  not   void  for  uncer- 
tainty, since  the  land  could  be  laid  off  in  a  strip  of 
equal  width  off  the  east  side  of  the  quarter  section, 
though  such  strip  included  part  of  the  lake. 

Mendota  Club  v.  Anderson   (Wis.  1899)   78  N.  W.  185. 

36.  Where  two  deeds   in  plaintiff's   chain  of   title 
respectively  define  the  boundary,  of  the  land  "  by  the 
edge  of  the   mill-pond "   and    as   "  the  bank   of   said 
mill-pond,"  and  defendant  is  entitled  to  pond  as  much 
land  as  the  pond  flowed  at  the  time  of  his  purchase, 
defendant  may  enter  on  land  orginally  covered  by  the 
pond,  but  which  has  subsequently  become  dry  land  by 
the  receding  of  the  water,  though  plaintiff's  deed  on 
its  face  shows  his  line  to  be  the  center  of  the  pond. 

Holden  v.  Chandler   (Vt.)   18  A.   310. 

37.  Where  the  patentee  of  "  the  north  half  of  the 
southeast    quarter,    and    that    part    of    the    northeast 
fractional  quarter,   of   Section  36,"   etc.,   "  which  lies 
north  of  the  Kankakee  river,  containing  in  all  122.70 
acres,"  conveys  "  the  northeast  quarter  of  Section  36," 


LANDS    BORDERING    ON    WATERS.  315 

etc.,  "  containing  122.70  acres,"  the  deed  passes  title 
to  all  of  the  land  in  said  northeast  fractional  quarter 
lying  south  of  said  river. 

Sphung  v.  Moore  (Ind.)  22  N.  E.  319. 

38.  Where  one  who  owns  a  tract  of  land  that  sur- 
rounds and  underlies  a  non-navigable  lake,  the  length 
of  which  is  distinguishably  greater  than  its  breadth, 
conveys  a  parcel  thereof  that  borders  on  the  lake,  by 
a  description  which  makes  the  lake  one  of  its  boun- 
daries, the  presumption  is  that  the  parties  do  not  in- 
tend that  the  grantor  should  retain  the  title  to  the  land 
between  the  edge  of  the  water  and  the  center  of  the 
lake,   and  the  title  of   the  purchaser,   therefore,   will 
extend  to  the  center  thereof. 

Lembeck  v".   Nye   (Ohio)   24  N.   E.  686. 

39.  A  patent  from  the  United  States  of  a  surveyed 
fractional  government  subdivision,  bounded  on  a  me- 
andered lake,  conveys  the  land  to  the  lake,  although 
the  meander  line  of  the  survey  be  found  to  be  not 
coincident  with  the  shore  line. 

Everson  v.  City  of  Waseca  (Minn.)  46  N.  W.  405. 

40.  When  the  United  States  has  disposed  of  the 
lands  bordering  on  a  meandered  lake,  by  patent,  with- 
out reservation  or  restriction,  it  has  nothing  left  to 
convey,  and  any  patent  thereafter  issued  for  land  form- 
ing the  bed,  or  former  bed  of  the  lake,  is  void  and  in- 
operative. Lamphrey  v.  Metcalf  (Minn.)  53  N.  W.  1139. 

41.  Where  the  United  States  has  made  grants  with- 
out reservation  or  restriction  of  public  lands  bounded 
on  streams  or  other  waters,  the  question  whether  the 
lands  forming  the  beds  of  the  waters  belong  to   the 
state,  or  to  the  owners  of  the  riparian  lands,  is  to  be 
determined  entirely  by  the  law  of  the  state  in  which 
the  lands  lie. 

Lamprey  v.  Metcalf  (Minn.)  53  N.  W.  1139. 


316  A   MANUAL   OF   LAND   SURVEYING. 

42.  Where  a  section  is  divided  by  a  water  course, 
and  is  subdivided  in  lots  instead  of  regular  subdivi- 
sions, and  a  lot  bounds  on  the  water  course,  the  water 
course  itself,  and  not  the  meander  lines  thereof,  is  the 
proper  boundary;    and,   if  the  grantee   does  not   find 
the  water  course  as  called  for  by  his  patent,  he  may 
go   as  far   as  the  next   "  eighth  line "    to  locate   his 
boundary. 

Lally  v.   Rossman(  Wis.)   15  N.  W.  1132. 

43.  Where  the  description  is  by  metes  and  bounds, 
no   reference   being   made    therein   to   the    lake,   then 
only  the  land  included  within  the  lines   as  fixed  by 
the  terms  used  by  the  parties  to  the  deed  will  pass  to 
the  grantee, 

Lembeck   v.    Nye    (Ohio)    24   N.    E.   686. 

44.  If,  however,  the  call  in  the  description  be  to 
and  thence  along  the  margin  of  the  lake,  no  such  pre- 
sumption arises,  and  the   title  of  the  purchaser  will 
extend  to  low  water  mark  only. 

Lembeck   v.    Nye    (Ohio)    24   N.    E.   686. 

45.  Where  a  deed  conveys  land  "  bounded  and  des- 
cribed according  to "   a  certain   survey,  does  not  call 
for  a  river,  but  calls  for  a  line  run  between  certain 
points,  designated  by  the  surveyor  as  on  the  bank  of 
&  navigable    river,   and   it   appears   that  the   lines   of 
such  survey  exclude  flats  between  high  and  low  water 
marks,  evidence  aliande   is  admissible  that  the  bank 
referred  to  was  an  artificial  dike ;  that  the  grantee  had 
notice  that  the  grantors  reserved  the  flats;  that  the 
grantors  refused  to  execute  a  deed  expressly  conveying 
the  flats ;  and  that  the  sale  was  expressly  subject  to  the 
survey,  as  tending  to  show  that  the  flats  were  excluded, 
whatever  may  be  the  presumption  from  the  deed. 

Palmer  v.  Farrell   (Pa.)   18  A.  761. 


GENERAL   RULES.  317 

Second. 

4.     In  locating  the  corners  and  boundary  lines  on 
the  ground,  we  will  consider: 

1.  General  rules  which  apply  to  all  resurveys : 

2.  Special   application   of  these  rules  to  the  rect- 
angular system  of  United  Stat.es  surveys. 

GENERAL   RULES. 

KULE  1. — In  locating  a  deed  on  the  ground,  we  are 
to  rely— 

(1)  On  the  actual  lines   originally  surveyed; 

(2)  On   lines    run    from    acknowledged    calls    and 
corners. 

(3)  On  lines  run  according  to  the  course  and  dis- 
tance in  the  deed. 

Avery  v.   Baum,  Wright's  Ohio,   576. 
1  Rich.    (S.   C.)    491. 

2.  When  the  boundaries  of  lands  are  fixed,  known 
and     unquestionable      monuments,      though      neither 
courses,  distances,  nor  computed  contents  correspond, 
the  monuments  must  govern. 

Pernam  v.  Wead,  6  Mass.  131. 

Nelson  v.  Hall,  1  McLean  (U.  S.)   518. 

3.  Though    known    and    fixed    monuments    control 
where  they   conflict   with   the   courses   and  distances, 
yet  where  there  are  two  conflicting  monuments,  only 
one  of  which  corresponds  with  the  courses  and   dis- 
tances, that  one  should  be  taken,  and   the  other  re- 
jected as  surplusage. 

Zeibold  v.   Foster  (Mo.   Sup.)  24  S.  W.  155. 

4.  While  natural  objects  usually  control  courses  and 
distances  in  boundaries  to  land,  the  rule  will  not  be 
applied    where    the    natural    object    is    shown    to    be 
variable  in  its  position. 

Smith  v.   Hutchinson    (Tenn.   1900)   58  S.  W.  226. 


318  A    MANUAL    OF    LAND    SURVEYING. 

5.  A  boundary  line  described  by  measurement  and 
without   monuments    will    govern,    although    the    dis- 
tance be  described  as  so  many  feet,  more  or  less. 

Adkins  v.    Quest    (Mo.   App.    1899)    79   Mo.   App.   36,   2   Mo.   App. 
Rep.    348. 

6.  Marked   lines   and  corners   control   courses  and 
distances.     Surplus  lands  do  not  vitiate  a  survey  nor 
does  a  deficiency  of  acres  called  for  in  a  survey  operate 
against   it.     Wherever   the   boundaries    can   be   estab- 
lished, they  must  prevail. 

Robinson  v.  Moore,  4  McLean     (U.  S.  C.  C.)   279. 
Morrow  v.    Whitney,   5   Otto    (U.    S.)    551. 

7.  A  deed  called  for  posts  as  corners.     The  survey 
was  made  and  the  posts  set  prior  to  the  execution  oi' 
the  deed.     It  was   afterward  found  that  there  was  a 
shortage  of  several  acres.     Held  that  proof  that  posts 
were  set  up  as  corners  between  adjoining  owners  con- 
trols the  call  for  course  and  distance. 

Alseire  v.  Hulse,  5  Ohio,  534. 

8.  The  rule  that  courses,  distances  and  quantities 
must  yield  to  monuments,  is  not  inflexible,  especially 
when  the  distances  are  very  short,  and  the  monuments 
artificial  ones,  as  here,  a  mill-race,  etc. 

Higinbotham  v.  Stoddard,  72  N.   Y.  94. 
Ga.  R.  R.  Co.  v.  Hamilton,  59  Ga.  171. 

9.  In  a  case  where  no  mistake  could  be  reasonably 
supposed  in  the  courses  and  distances,  the  reasons  of 
the  rule  were  held  to  fail,  and  the  rule  was  not  ap- 
plied. 

Davis  v.  Rainsford,   17  Mass.  207. 

10.  The  rule  that  natural  or  artificial  boundaries 
will  control  distances  or  courses,  authorizes  no  other 
departure  from  the  course  or  distance  than  such  as  is 
necessary    to    effectuate   the   apparent    intent    of   the 
grantor. 


GENERAL    RULES.  319 

Distances  may  be  increased  and  courses  departed 
from  in  order  to  preserve  the  boundary,  but  the  rule 
authorizes  no  other  departure  from  the  course  and  dis- 
tance than  such  as  is  necessary  to  preserve  the  bound- 
ary. 

Johnson  v.  McMillan,  1  Strobh.   (S.  C.)   143. 

11.  If  the  courses  and  distances  cannot  be  other- 
wise reconciled  with  the  monuments  in  a  description, 
a  line  in  a  survey  which  has  evidently  been  omitted 
will  be  supplied  to  prevent  the  obvious  intent  of  the 
grantor   from   being  frustrated.     Serrano   v.    Rawson. 
47    Cal.    52.     See   also    Schultz   v.   Young,   3   Iredell, 
N.   C.,  385,  where  two  lines  must  be  run  instead  of  the 
one   called  for,  to  best   conform  with  the  whole  de- 
scription in  the  deed. 

12.  A  survey  must  be  closed  in  some  way  or  other. 
If  this  can  only  be  done  by  following  the  course  the 
proper    distance,    then   it    would    seem    that    distance 
should  prevail;  but  when  the  distance  falls  short  of 
closing,  and  the  course  will  do  it,  the  reason  for  ob- 
serving distance   fails. 

Doe  v.  King,  3  How.  Miss.  125. 

13.  Where  land  conveyed  forms  a  triangle,  and  two 
sides  and  the  acreage  are  given,  a  straight  line  from 
point  to  point  will  be  adopted  as  the  third  side,  when 
the  boundary  thus  formed  will  enclose  the  number  of 
acres  called  for. 

Hostetter  v.  Los  Angeles  Terminal  Ry.  Co.   (Cal.)  41  P.  330. 

14.  Where  three  sides  and  the  number  of  acres  are 
known,  and  it  is  disputed  whether  the  fourth  side  is  a 
straight  or  meandering  line,  the  straight  line  will  be 
adopted,   when   the   tract   thus   enclosed   contains   the 
number  of  acres   called   for,   and   when   the   acreage 


320  A   MANUAL    OF    LAND    SURVEYING. 

\vould  be  largely  increased  if  the  meandering  line  were 
adopted. 

Hostetter  v.   Los  Angeles  Terminal  Ry.  Co.   (Cal.)   41  P.  330. 

15.  It  is  a  universal  rule  that  course  and  distance 
yield  to  natural  and  ascertained  objects.     But  where 
these  objects  are  wanting,  and  the  course  and  distance 
can  not  be  'reconciled,  there  is  no  universal  rule  that 
obliges  us  to  prefer  the  one  to  the  other.     Cases  may 
exist  in  which  either  one  may  be  preferred,  according 
to  the  circumstances. 

Preston's  Heirs  v.  Bowman,  6  Wall.   (U.  S.)  580. 

16.  If  no  principle  of  location  be  violated  by  clos- 
ing from  either  of  two  points,  that  may  be  closed  from 
which  will  be  more  against  the  grantor  and   include 
the  greater  quantity  of  land. 

Johnson  v.  McMillan,  1  Strobh.   (S.  C.)   143. 

17.  The  boundary  line  is  to  be  ascertained  by  run- 
ning direct  lines  from  one  monument  to  the  other. 

Melcher  v.  Merryman,  4  Me.  601. 

18.  A  line   actually  marked   must  be   adhered   to, 
though  not  a  right  line  from  corner  to  corner.     Where 
a  line  has  been  marked  only  part  of  the  way,  the  re- 
mainder  of  the   line  must   run   direct  to   the  corner 
called  for. 

Cowan  v.   Fauntleroy,  2  Bibb   (Ky.)   261. 

19.  A  marked  line   of   another  tract,  when  called 
for  in  a  conveyance,   must  be  run  disregarding   dis- 
tance; but  where  such  line  can  not  be  established,  the 
distance  run  must  govern. 

Cause  v.   Perkins,  2  Jones  Law  Rep.    (N.   Y.)   222. 

20.  Where  a  line  is  described  as  running  a  certain 
distance  to  a  particular  monument,   and  that  monu- 
ment has  disappeared  and  its  place   cannot  be  ascer- 
tained, the  course  and  distance,  in  the  absence  of  other 
controlling  words,  must  govern. 

Budd  v.   Brooke,  3   Gill    (S.   C.)    198. 

See  also  Bruckner  v.  Lawrence,  I  Douglass  (Mich.)  19. 


GENERAL    RULES.  321 

21.  Course  and  distance  yield  to  known,  visible  and 
definite  objects;  but  they  do  not  yield  unless  to  calls 
more  material  and  equally  certain. 

Shipp  et.  al.  v.  Miller's  Heirs,  2  Wheat.   (U.  S.)  316. 

22.  Courses  and  distances  in  the  deed  are  not  to  be 
controlled  by  monuments  or  objects  variant  therefrom 
and  not  called  for  in  the  description,  but  they  must 
yield  to   such  objects   and   monuments   as   are  refer- 
red to. 

Bruckner's   Lessee  v.   Lawrence,  1    Doug.,   Mich.   29. 
Moore   v.    People,    2    Doug.,  Mich.    424. 
Bower  v.  Earle,  18  Mich.  165. 

23.  Wherever   it  can  be  proved  that  the  line  was 
actually  run,  was  marked,  and  the  corners  made,  the 
party  claiming  under  the  deed  will  hold  accordingly, 
although  there  is  a  mistake  in  the  description  in  the 
deed. 

Cherry  v.   Slade,  3  Murph,    (N.   C.)   82. 

24.  A  sold  to  B  lot  7,  informing  B,  at  the  time  of 
the  sale,  that  it  was  four  rods  wide,  and  marking  it 
out  upon  the  ground.     He  subsequently  sold  to  C  lot 
8  and  a  vacated  alley  one  rod  in  width  between  lots 
7  and  8,  informing  C,  at  the  time,  that  lot  8  was  four 
rods  wide,   and  the  alley  one  rod   wide,  making  five 
rods  in  all,  and  pointing  out  to  C  the  marks  previously 
made  by  him  for  the  boundary  of  lot  7,  sold  to  B,  as 
being  also  the  boundary  of  the  alley  sold  to  C.     The 
premises   were   occupied   by  B   and   C   in   accordance 
therewith,     without     dispute.      It     was     subsequently 
found,  by  reference  to  the  plat,  that  lot  7  was  five 
rods  wide,  and  that  there  was  no  alley  between   the 
lots;  whereupon  B  claimed  the  additional  rod.     Held, 
that  to  allow  B  to  hold  the  rod  in  width  of  land  which 
she  did  not  purchase  or  pay  for,  and  to  deprive  C  of 

21 


322  A    MANUAL    OF    LAND    SURVEYING. 

land  x which  he  did   purchase  and  pay  for,  would  be 
both  bad  law  and  bad  morals. 

Bolton  v.   Eggleston,  Iowa. 
N.  W.  Rep.,  Vol.  16,  P.  62. 

25.  Boundary    may    be    proved    by    any    evidence 
which  is  admissible  to  establish  any_  other  fact. 

Smith  v.  Prewitt,  2  A.  K.  Marsh.   (Ky.)  158. 

26.  Where  no  bounds  were  established,  the  dividing 
line  must  be  run  by  aid  of  the  measurements  in  the 
deeds,  the  oldest  title  receiving  its  full  measure  first. 

Talbott  v.   Copeland,  38  Me.  333. 

27.  A  long  established  fence  is  better  evidence  of 
actual  boundaries,  settled  by  practical  location,  than 
any  survey  made  after  the  monuments  of  the  original 
survey  have  disappeared.     A  resurvey  made  after  the 
monuments  of  the  original  survey  have  disappeared,  is 
for  the  purpose  of  determining  where  they  were,  and 
not  where  they  ought  to  have  been. 

Diehl  v.   Zauger,  39  Mich.   601. 

Hunt's  Lessee  v.  McHenry  and  Williams,  Wright's   (Ohio)   599. 

28.  Where  between  the  plan  and  the  original  survey 
there  is  a  difference  in  the  location  of  the  lines  and 
monuments,     the     lines     and     monuments     originally 
marked  as  such  are  to  govern,  however  much  they  may 
differ  from  those  represented  on  the  plan. 

Ripley  v.   Barry,  5  Greenl.    (Me.)   24. 

See  also  2  Greenl.   (Me.)  214,  and  3  Gr.    (Me.)   126. 

29.  But  no  such  rule  has  obtained  where  the  survey 
was  subsequent  to  the  plan. 

Thomas  v.   Patten,   1   Shep.    (Me.)    329. 

30.  Purchasers  of  town  lots  have  a  right  to  locate 
them  according  to  the  stakes  which  they  find  planted 
and    recognized,    and    no    subsequent    survey    can    be 
allowed  to  unsettle  them.     The  question  afterwards  is 
not  where  they  should  have  been,,  in  order  to  make 


GENERAL    RULES.  323 

them  correspond  with  the  lot  lines  as  they  should  be 
if  the  platting  were  done  with  mathematical  accuracy, 
but  it  is  whether  they  were  planted  by  authority,  and 
the  lots  were  purchased  and  taken  possession  of  in 
reliance  on  them.  If  such  was  the  case,  they  must 
govern,  notwithstanding  any  errors  in  locating  them. 

Flynn  v.  Glenny,  51  Mich.  580. 

31.  In  ascertaining  the  true  line  of  a  city  street, 
fences  built  by  adjoining  lot  owners  on  the  line  of  the 
street,  according  to  stakes   set  by  the  surveyor  soon 
after  the  original  survey  was  made,  and  maintained 
for  45  years,  are  better  evidence  of  the  location  of  such 
line   than   a   new   survey,    made   40   years    after   the 
original  survey,  which  changes  such  line. 

City  of  Racine  v.  Emerson  (Wis.)  55  N.  W.  177. 

32.  Of  two  overlapping  surveys,  the  one  first  made 
has  priority,  particularly  where  the  second  is  bounded 
with  express  reference  to  the  first. 

Van  Amburgh  v.  Hitt  (Mo.  Sup.)  22  S.  W.  636. 

33.  Any  calls  of  the  second  survey  conflicting  with 
monuments  and  calls  of  the  first  must  yield  thereto. 

Van  Amburgh  v.  Hitt  (Mo.  Sup.)  22  S.  W.  636. 

34.  Where  two  surveys  call  for  each  other,  there 
can  be  no  vacancy   unless  the  lines  marked   on   the 
ground    contradict    the    call;    and    in    such   case   the 
marked  lines  must  govern. 

McGinnis  v.  Porter,  20  Penn.  80. 

35.  Where   two    surveys   made    twenty-three   years 
apart  are  found  to  disagree,  the  probabilities  favor  the 
earlier    survey    when    the   original    corners    and    wit- 
nesses are  gone  at  the  time  of  the  last  survey,  espe- 
cially if  the  line  of  the  first  survey  has  remained  un- 
questioned for  many  years. 

Case  v.  Trapp,  49  Mich.  61. 


324  A    MANUAL    OF    LAND    SURVEYING. 

36.  When  the  same  grantor  conveyed  to  two  per- 
sons, to   each  one  a  lot  of  land,   limiting  each   to  a 
certain  number  of  rods  from  opposite  known  bounds 
running  in  a  direction  to  'meet  if  extended  far  enough, 
and  by  measure  the  lots  do  not  join — when  it  appears 
from  the  same  deeds  that  it  was  the   intention  that 
they  should  join,  a  rule  should  be  applied  which  will 
divide  the  surplus  between  the  grantees  in  proportion 
to  the  length  of  the  respective  lines  as  stated  in  their 
deeds. 

Lincoln  v.    Edgecomb,   28   Maine,   275. 

37.  Where  original   surveys  have   been  made,   and 
returned  as  a  block  into  the  land  office,  the  location  of 
each  tract  therein  may  be  proved  by  proving  the  loca- 
tion of  the  block.     In  ascertaining  the  location   of  a 
tract,  the  inquiry  is  not  where  it  should  or  might  have 
been  located,  but  where  it  actually  was  located. 

38.  Every  mark  on  the  ground  tending  to  show  the 
location  of  any  tract  in  the  block,   is  some  evidence 
of  the  location  of  the  whole  block,  and   therefore  of 
each  tract  therein. 

Coal  Co.  v.  Clement,  95  Pa.  St.  126.. 

39.  The  beginning  corner  of  a  survey,  as  given  in 
the  field  notes,  is  of  no  more  dignity  than  any  other 
corner  found  on  the  ground. 

Cox  v.  Finks  (Tex.  Civ.  App.)   41  S.  W.  95. 

40.  Where  lots  are  conveyed  by  number  according 
to   a  plat  which  is  made  from  an  actual  survey,  the 
corners  and  lines  fixed  by  that  survey  are  to  be  re- 
spected. 

Pyke  v.   Dyke,   2   Greenl.   Me.    214. 

41.  Streets  which  are  well  defined,  and  designated 
by  some  natural  or  artificial  monument,  must  govern 
course  and  distance  in  fixing  boundaries  of  lands;  but 
streets  which  are  not  thus  defined,  and  themselves  re- 


GENERAL    RULES.  325 

quire   to    be   located,    would    furnish    very   uncertain 
guides  in  arriving  at  the  boundaries  of  other  lands. 

Saltenstall  v.  Riley,  28  Ala.  164. 

42.  When  streets  have  been  opened  and  long  ac- 
quiesced in,  in  supposed  conformity  to  the  plat,  they 
should  be  accepted  as  fixed  monuments  in  locating  lots 
or  blocks  contiguous  thereto  or  fronting  thereon. 

Van  den  Brooks    v.  Correon,  48  Mich.  283. 

43.  Lands  have  been  laid  off  into  lots  and  blocks, 
and  platted,  before  being  cleared,  when,  by  reason  of 
inequalities  of  the  surface,   logs,  and   other  obstruc- 
tions, strictly  accurate  surveys  were  not  and  could  not 
be  made.     Where  the  blocks  and  streets  were  staked 
out  at  the  time,  such  monuments  would  be  fixed  and 
permanent,  leaving  the  excess  or  shortage  to  be  dealt 
with  by   itself.     So   where   the   streets,   although  not 
so   designated,   have  by  the  parties   interested   or  by 
the  public  authorities  been  opened,  used,  and  acqui- 
esced in,  they  thereby  become  permanent  boundaries 
and  form  new  starting  points  in  subsequent  surveys 
of  the  premises. 

Twogood  v.  Hoyt,  42  Mich,  609. 

44.  -Ancient  reputation  and  possession  in  regard  to 
streets  in  a  town  are  entitled  to  more  respect  in  de- 
ciding  on   the   boundaries   of   lots   than   any   experi- 
mental survey  that  may  be  afterwards  made. 

Ralston  v.  Miller,  3  Rand.   (Va.)   44. 

45.  Where  lots  are  sold  by  numbers  and  a  plat,  any 
variance   in    the   distance    between   known    and   fixed 
points   as   found  by    actual   measure   on   the   ground, 
and  the  distance  between  the  same  points  as  laid  down 
on  the  plat,  is  to  be  divided  between  the  lots  in  pro- 
portion to  the  respective  lengths  as  laid  down  on  the 
plat. 

Marsh  v.  Stephenson,  7  Ohio,  N.  3,  264. 
Quinnin  v.  Reimers,  46  Mich,  605. 


326  A    MANUAL    OF    LAND    PURVEYING. 

46.  Surplus  or  shortage  'in  a  block  is  to  be  divided 
pro  rat  a  between  the  lots. 

Newcomb  v.   Lewis,  31   Iowa     488. 
O'Brien  v.  McGraw,  27  Wis.  446. 

47.  Where  the  accuracy  of  the  starting  points  taken 
for  test  surveys  is  merely  matter  of  speculation,  they 
cannot  be  used  to   fix  a  disputed   boundary  between 
two  lots  when  the  dispute  arises  from  a  discrepancy 
which  affects  all  the  lots  in  a  block,  and  must  there- 
fore be  apportioned  among  them. 

Reimers  v.  Quinnin,  49  Mich.  449. 

48.  A  resurvey  is  inadmissible  in  evidence  to  show 
that  a  private  boundary   is  incorrect,   if  its   starting 
point  is  outside  of  and  does  not  belong  to  the  immedi- 
ate plan  or  local  system  by  which  the  original  survey 
was  controlled. 

Burns  v.  Martin,  45  Mich.  22. 

49.  If  in  running  the  lines  of  the  grant,  one  line  be 
found  which  is  admitted  or  proved  to  be  a  line  of  the 
grant,  which  will  run  with  a  variation  from  the  calls 
of  the  grant,  if  no  other  marked  lines  be  found,  the 
other  calls  should  be  run  with  the  same  variation  as 
that  found  on  the  marked  line. 

Sevier  v.   Wilson,   Peck.    146. 

50.  Where  a  deed  convey 3  lots  in  a  town,  and  refers 
to  a  plat  to  identify  them,  and,   in  describing  their 
lines,  calls  the  points  of  compass  as  designated  on  the 
plat  by  its  lines  and  angles,  a  correct  survey  cannot  be 
based  on   any  other  system;   and   although  the  lines 
there    delineated    are    not    comformable    to    the    true 
meridian,  the  plat  and  not  the  compass  should  govern. 

Bower  v.   Earl,  18  Mich.  367. 

51.  An  instruction  that,  in  arriving  at  a  boundary 
line  as  originally  run,  natural  objects  are  controlling 
calls;  artificial  objects,  second  in  importance;  course, 


GENERAL    RULES.  327 

third,  and  distance,  fourth;  and  that,  where  there  is 
still  uncertainty,  that  rule  should  be  adopted  most 
consistent  with  the  intent  of  the  grant,  is  correct. 

Luckett  v.  Scruggs  (Tex.)  11  S.  W.  529. 

52.  An  instruction  that  the  beginning  corner  of  a 
survey  is  of  no   higher   dignity    or   importance   than 
any  other  corner,  and  that,  "  if  there  are  well-known 
and  undisputed  original  corners  established  upon  the 
ground   around   the   survey,   they   would    control   the 
other  calls  of  the  survey,  which  are   conflicting  and 
contradictory,  if  there  are  any  such,"  is  correct. 

Luckett  v.  Scruggs  (Tex.)  11  S.  W.  529. 

53.  Where  the  beginning  corner  of  a  survey  is  the 
southwest,   but   the   southeast   corner  is   equally  well 
identified,  a  charge  limiting  the  jury  to  finding  the 
unidentified  northeast  corner  by  the  first  and  second 
lines  from  the  southwest  corner,  is  erroneous,  as  the 
southeast  corner  is  of  equal  importance,  unless  the  line 
from  the  former  corner  was  actually  run  and  measured, 
and  that  from  the  latter  not. 

Scott  v.  Pettigrew  (Tex.)  12  S.  W.  161. 
Lancaster  v.  Ayres,  Id.  163. 

54  An  instruction  making  the  importance  of  an 
established  northeast  corner,  in  locating  the  north  and 
west  lines  of  a  survey,  dependent  upon  the  jury's 
belief  that  such  western  line  was  not  run,  is  erroneous, 
as  such  corner  has  the  same  weight  for  the  purpose  in 
question,  whether  the  western  line  was  run  or  not. 

Scott  v.  Pettigrew   (Tex.)   12   S.  W.   161. 

55.  In  the  description  of  lands,  as  to  questions  of 
boundaries  the  rule  is  settled  in  Virginia  and  West 
Virginia  that  natural  land-marks,  marked  lines  and 
reputed  boundaries  will  control  mere  courses  and  dis- 
tances, or  mistaken  descriptions  in  surveys  and  con- 
veyances. 

Gwynn  v.  Schwartz  (W.  Va.)  9  S.  E.  880. 


328  A    MANUAL    OF    LAND    SURVEYING. 

56.  The  course  of  the  eastern  line  of  the  H.  tract, 
as  given  in  the  original  survey  made  in  1745,  was  14 
deg.  east.     The  course  of  the  western  line  of  the  B. 
tract,  lying  immediately  east  of  the  H.  tract,  as  given 
in  the  original  survey  made  in  1813,  was  17  deg.  and 
15  min.  east.     The  western  line  of  the  B.  tract  was 
made  of  exactly  the  same  length  as  the  eastern  line  of 
the  H.  tract,  and  the  beginning  point  of  the  two  lines 
was  the  same.     The  difference  in  the  course  of  the  two 
lines  could  be  satisfactorily  explained  by  the  change 
in   the   position    of   the   magnetic    needle    which    had 
taken  place  in  the  time  intervening  between  1745  and 
1813.     Held,  that  the  two  lines  must  be  considered  as 
coincident. 

Scott  v.  Yard  (K.  J.)  18  A.  359. 

57.  Where   neither   the    corners   of   plaintiffs'   nor 
defendants'    land    are    satisfactorily    established,    and 
there    is   a   well-established    and    identified   corner   of 
another  survey,  from  which,  by  following  course  and 
distance,  defendants'  survey  can  be  constructed,  such 
course  should  be  followed  though  the  boundaries  thus 
established    include    land    within    the    boundaries    of 
plaintiffs'  junior  survey. 

Griffith  v.  Rife  (Tex.)  12  S.  W.  168. 

58.  A    county    surveyor,    employed   to    restore    the 
lines  and  corners  of  adjoining  tracts  of  land  accord- 
ing to  the  original  government  survey,  found  township 
corners  only,  then  (the  other  quarter  and  section  cor- 
ners being  missing)  ran  a  straight  line  from  one  town- 
ship corner  to  the  other,  and  on  this  line  placed  the 
quarter  and  section  corners,  but  did  not  take  any  testi- 
mony to  ascertain  the  lines  or  corners  of  the  original 
survey,  did  not  attempt  to  prove  his  lines  or  corners 
by   re-establishing  the   missing    corners   from   all   the 
nearest  known  original  corners,  in  all  directions,  did 


GENERAL    RULES.  32S 

not  sufficiently  regard  the  field  notes,  and  did  not, 
where  the  original  monuments  had  disappeared,  regard 
the  boundary  lines  long  recognized  and  acquiesced  in. 
Held,  that  such  a  survey  is  incomplete,  and  cannot 
be  approved  as  the  true  and  correct  determination  of 
the  boundaries  and  corners  as  originally  established  by 
the  government.  ;  ^  , 

Reinert  v.  Brunt  (Kan.)  21  P.  807. 

59.  Upon  an  issue  as  to  the  location  of  a  line  of  the 
government  survey,  evidence  of  the  location  of  monu- 
ments is  not  overcome  by  field-notes  of  the  original 
survey,  taken  at  the  time  of  the  erection  of  said  monu- 
ments or  subsequent  thereto. 

Hubbard   v.    Dusy    (Cal.)    22    P.    214. 

60.  As  between  complicated  descriptions  of  a  line 
dividing  two  sections  or  quarter  sections,  that  one  is 
to  be  adopted  which  is  most  in  conformity  with  the 
monument  established  by  the  government  survey. 

Hubbard   v.    Dusy    (Cal.)    22    P.    214. 

61.  As    between   different    monuments,    those   best 
identified  should  prevail,  independent  of  anything  in 
the  field-notes  of  the  original  or  any  subsequent  sur- 
vey. 

Hubbard   v.    Dusy    (Cal.)    22    P.    214. 

62.  Where   it   is  x doubtful   which   of   two   lines   of 
monuments  is  the  true  government  line,  other  things 
being  equal,   that  one  is  to  be   so   considered   which 
most  nearly  conforms  to  the  field-notes. 

Hubbard   v.    Dusy    (Cal.)    22    P.    214. 

63.  Where,  in  ejectment,  the  location  of  the  bound- 
ary line  between  two  lots  is  in  question,  and  the  lots 
were  staked  when  platted,  such  monuments  are  con- 
clusive of  the  question;  but  if  they  were  not  staked, 
other   monuments,    establishing    any   given   points   as 
platted,  furnish  starting  points  to  aid  in  arriving  at 


330  A    MANUAL    OF    LAND    SURVEYING, 

the  true  boundary,  and,  in  the  absence  of  either,  old 
monuments  indicating  user  may  .be  resorted  to. 

Brudin  v.   Inglis   (Mich.   1899)   80  N.   W.   115. 

64.  Where  there  is  a  discrepancy,  in  a  government 
survey,  between  the  monuments  and  the  distances 
given  in  the  field  notes,  the  monuments  will  control, 
event  though  the  result  be  that  some  of  the  quarter 
sections  will  contain  less  than  their  proper  number  of 
acres. 

Ogilvie  v.   Copeland   (111.   Sup.)   33  N.  E.  1085. 

05  In  the  rule  that  monuments  control  courses  and 
distances,  and  that  when  monuments  and  measure- 
ments vary,  the  monuments  always  control,  the  refer- 
ence is  to  monuments  and  measurements  made  by  the 
original  survey. 

Woodbury  v.  Venia   (Mich.   1897)   72  N.   W.  189. 

66.  On  a  question  as  to  the  true  location  of  a  land 
patent,  boundaries  fixed  by  reversing  the  courses  and 
distances  must  govern  when   found  to  coincide  with 
the  natural  calls  of  the  patent. 

Ellinwood  v.   Stancliff,  42   F.  316. 

67.  When  the  points  fixed  by  reversing  the  courses 
and  distances  do  not  coincide  with  the  natural  calls 
of  the  patent,  or  the  natural  calls  cannot  be  identified, 
then  the  regular  courses  and  distances  must  govern. 

Ellinwood  v.   Stancliff,  42   F.  316.' 

68.  When    a    survey    calls    for    the    "  Dougherty " 
survey  as  one  of  its  adjoiners,  an  instruction  that  if 
the  jury  find  that  the  "  King  "  is  the  survey  intended 
by  the  call  for  "  Dougherty,"  the  former  being  located, 
the  call  would  furnish  "  some  evidence  "  of  the  loca- 
tion of  the  survey  in  question,  is  insufficient,  as  such 
a  finding  would  locate  the  survey  in  the  absence  of 
marks  upon  the  ground. 

Tyrone  Min.  &  Manuf'g    Co.  v.   Cross   (Pa.)   18  A.  519. 


GENERAL    RULES.  331 

69.  Where  no  marks  are  found  on  the  boundaries 
of  a  survey,  and  it  cannot  be  located  on  the  ground, 
evidence  of  the  location  of  junior  surveys  which  call 
for  the  lines  of  the  elder  as  adjoiners  is  admissible, 
as    showing    where    the    surveyors    upon    the    ground 
located  such  lines. 

Tyrone  Min.  &  Manuf'g    Co.  v.  Cross   (Pa.)   18  A.  519. 

70.  Where    the   distances   of   a    survey   have    been 
actually  measured  upon  the  ground,  the  courses  and 
distances  may  be  reversed  when  by  so  doing  they  more 
nearly  harmonize  with  the  natural  calls  of  the  patent, 
and  the  "  beginning "   corner   does   not  control  more 
than  any  other  corner  which  is  definitely  ascertained. 

Ayers  v.  Watson,  11  S.  Ct.  201. 

71.  Where  the  court,  in  an  action  of  ejectment,  in- 
structs the  jury  that,  "  after  a  survey  of  blocks  had 
been  returned  and  had  remained  in  the  land-office  21 
years,  it  was  conclusively  presumed  that  it  was  run 
upon  the  ground,  whether  marks  were  found  upon  the 
ground  or  not,"  but  in  other  portions  of  his  charge 
repeatedly  states  the  law  to  be  that  marks  made  by 
the  surveyor  on  the  ground  are  the  first  and  highest 
evidence  of  the  true  survey,  the  instruction  cannot,  on 
the  whole,  be  said  to  be  misleading,  as  he  will  be  rea- 
sonably understood  to  have  charged  that  the  presump- 
tion in  favor  of  returns  of  surveys  on  file  for  21  years 
is   only   applicable  to   such  surveys   where  no   monu- 
ments or  marks  on  the  ground  are  found  to  contradict 
them. 

Grier  v.  Pennsylvania  Coal  Co.   (Pa.)  18  A.  480. 

72.  The  exterior  of  two  adjoining  interior  surveys 
were   undisputed.     The   boundary   line   between   them 
had  never  been  surveyed,   but   its  southern   end  was 
marked  by  an  oak.     North  of  these  surveys  were  two 
others.     These  four  surveys  were  originally  returned 


332  A    MANUAL    OF    LAND    SURVEYING. 

as  being  of  equal  size,  and  having  one  common  corner. 
The  northern  end  of  the  line  between  these  two  latter 
surveys  was  marked  by  a  sugar-maple;  which  was  not 
directly  opposite  the  oak,  and  it  was  proved  that  the 
northern  line  of  these  surveys  was  shorter  than  the 
southern  line  of  the  others.  Held,  that  the  boundary 
line  between  the  two  southern  surveys  should  run  from 
the  oak  parallel  to  the  end  lines,  and  not  diagonally 
from  the  oak  to  the  maple. 

Bloom  v.   Ferguson    (Pa.)    18  A.    488. 

73.  Where   a  dividing  line  is   established   between 
tracts  of  land  owned  by  a  county,   before   purchases 
are  made  of  land  on  each  side  of  it,  and  the  deeds 
under  which  parties  claim  have  been  made,   and  are 
known  by  the  parties  to  have  been  made  with  refer- 
ence to  that  line,  they,  and  all  the  persons  claiming 
through  them,  are  bound  by  it. 

Briscoe  v.  Puckett  (Tex.)  12  S.  W.  978. 

74.  The  northwest  corner  of  a  survey  was  plainly 
marked,  and  part  of  the  west  line  was  also  marked. 
The  rest  of  the  survey  had  apparently  not  been  run  on 
the  ground,  but  the  southeast  corner  was  ascertainable 
from  the  field-notes,  being  located   on  an  established 
line  of  another  survey  and  at  a  given  distance  from 
an  established  point.     The  lines  of  survey  as  called  for 
in  the  field-notes  were  correct  as  to  courses  but  were 
too  short  to  reach  from  one   of  said  corners  to  the 
other.     Held,  that  the  survey  included  all  the  land  be- 
tween the  corners  bound  by  the  lines  as  extended  so  as 
to  reach  from  one  corner  to  the  other. 

Randall  v.  Gill   (Tex.)   14   S.  W.  134. 

75.  Where  a  deed . describes  a  lot  conveyed  as  of  a 
certain  width,  and  a  party-wall   stands  on  the  south 
line,  the  north  line  may  be  found  by  measuring  the 
given  distance  north  from  the  middle  of  such  wall. 

Warfel  v.  Knott  (Pa.)   18  A.  390. 


GENERAL    RULES.  333 

76.  The  statement  of  the  quantity  of  land  supposed 
to  be  conveyed,  and  inserted  in  deeds  by  way  of  de- 
scription, must  not  only  yield  to  natural  land-marks 
and  marked  lines,  but  also   to  descriptions  in  deeds 
by  courses  and  distances. 

Gwynn  v.  Schwartz  (W.  Va.)   9  S.  E.  880. 

77.  A  call  for  a  lot  by  the  name  or  number  which 
it  bears  on  a  plat  of  the  land  will  prevail  over  courses 
and   distances,   and   ordinarily   over   calls   for   monu- 
ments. 

O'Herrin  v.   Brooks    (Miss.)    6   So.   844. 

78.  Where  the  descriptions  in  a  deed  refer  to  a  sur- 
vey and  a  map  based  thereon,  making  both  a  part  of 
the  deed,  and  there  is  a  discrepancy  between  the  map 
and  the  survey,  the  latter  will  prevail. 

Whiting  v.  Gardner  (Cal.)  32  P.  71. 

79.  The  owner  of  a  lot  in  the  city  of  Rochester,  of 
the  area  of  about  one-half  acre,  rectangular  in  form, 
fronting  274  feet  on  a  street,  and  abutting  on  the  rear 
for  the  same  distance  on  a  canal,  the  location  of  both, 
as  well  as  the  other  lines,  being  undisputed,  conveyed 
a  portion,  by  description,  of  "  137  feet  front  and  rear, 
measuring  from  G.  H.'s  north  line  on  G.  street,  and 
also  137  feet  from  G.  H.'s  south  line  on  the  canal; 
being  the  piece  of  land  occupied  as  a  garden  by  the 
grantor."     The  lot  was  divided  by  a  'fence,  one  side 
being  used  as  a  garden ;  the  fence  starting  on  G.  street 
midway,  but  striking  the  back  line  at  the  canal  at  a 
point   19^   feet   from  the   middle   of   the   lot.     That 
fence  was  not  mentioned  in  the  deed.     Held,  that  the 
reference  to  the  garden  was  too  indefinite  to  control 
the  calls  for  exact  distances  from  known  bounds,  and 
the  divisional  point  on  the  canal  should  be  located  137 
feet  from  G.  H  's  line. 

Harris  v.  Oakley,  7  N.  Y.  S.  232. 


*i    .  • 


334  A    MANUAL    OF    LAND    SURVEYING. 

80.  Plaintiff  owned  a   village  lot,  No.   124,   and  a 
tract  of  land  lying  adjacent  thereto  on  the  south  and 
east    sides.     Eiver    street,    which   lay   along   a    river's 
edge,  was  the  westerly  front  of  both  the  lot  and  the 
tract.     He  conveyed  the  tract  to  defendant,  reserving 
a  part  thereof,  beginning  at  the  S.  W.  corner  of  the 
lot;  thence  southeasterly,  along  River  street,  32  feet; 
thence  northeasterly,   "  on   a   line   with   the   southeast 
corner  of  lot  No.  124,"  10  rods  and  23  links;  thence 
N.  to  M.  street;  thence  W.  to  the  N.  E.  corner  of  the 
lot;  thence  southwesterly,  to  the  S.  E.  corner;  thence 
to   the   beginning.     Locating   the  beginning  point    at 
the  S.  W.  corner  of  the  lot  as  appeared  by  the  village 
plat  on  the  easterly  side  of  the  street,  the  line  passed 
directly  through  the  S.  E.  corner  of  lot  124,  taking  no 
part    of    the    lot,    and    thus    making    the    reservation 
wholly- within  the  tract  conveyed;  but  by  beginning 
at  the  river's  edge,  on  the  westerly  side  of  the  street, 
on  the  theory  that  plaintiff's  property  extended  to  the 
river,  subject  only  to  the  easement  of  the  street,  the 
line   would   pass   through   and   take  part   of   lot   124. 
Held,  that  the  former  location  of  the  corner  w$s  cor- 
rect. 

Anderson  v.   Scott   (Mich.)   42  N.   W.  991. 

81.  In  an  action  to  recover  a  tract  of  land  lying 
between  a  slough  and  a  river,  plaintiff  claimed  title  by 
virtue  of  a  grant  which  bounded  the  land  granted  by 
the  river,  and  the  defendant  introduced  evidence  that 
the  surveyor  who  surveyed  the  grant  meandered  the 
slough  instead  of  the  river.     Held,  that,  in  determin- 
ing the  true  boundaries   of  the  grant,  the  sole  ques- 
tion was  to  ascertain  exactly  where  the  surveyor  ran 
his  lines,  and,  if  the,  jury  found  that  he  ran  the  line 
along  the  slough,  they  should  find  for  the  defendant. 

Allen  v.    Koepsel    (Tex.)    14   S.   W.   151. 


GENERAL    RULES.  335 

82.  Where,  in  ejectment,  a  surveyor  testified  that 
he  ran  the  boundary  line  in  dispute  about  1S68;  that 
he  found  the  original  stake  of  the  government  survey 
at  the  section  corner,  and  used  it  as  a  starting  point; 
and  it  appeared  that  about  the  same  time  defendant 
built  a  fence  upon  this  line,  which  he  has  ever  since 
maintained — this  line  must  prevail  over  one  surveyed 
20  years  later,  when  the  corner   mark  was  gone,  by 
one  who  testified  that  he  located  the  section  corner  by 
measurements  from  various  lines  and  points,  and  then 
by  digging  found  a   stump  which  he  took  to  be  the 
original  witness,  and  based  his  survey  upon  it. 

Carpenter  v.  Monks   (Mich.)   45  N.  W.  477. 

83.  The  monuments  or  marks  of  the  surveyor  on 
the  ground  determine  the  true  survey  as  against  calls 
for  adjoinders  or  courses  and  distances  as  returned; 
but,  each  block  of   surveys  being  separate  and  com- 
plete of  itself,  the  call  of  a  tract  in  one  block  for  an 
adjoinder  in  another  does  not  make  the  monument  of 
the  adjoinder  the  monument  of  the  later  block. 

Grier  v.  Pennsylvania  Coal  Co.   (Pa.)   18  A.  480. 

84.  Where  a  boundary  line  is   assented  to  by  the 
owner  of  a  tract  of  land  at  a  time  when  there  is  ni 
dispute  concerning  such  line,  and  on  the  supposition 
that  it  is  the  true  boundary,  he  is  not  estopped,  on 
discovering  that  such  is  not  the  case,  from  claiming 
title  to  the  real  boundary. 

Schraeder  Min.  &  Manuf'g  Co.  v.  Packer,  9  S.  Ct.  385. 

85.  Continuous      and      uninterrupted      possession, 
under  claim  of  ownership,  to  the  line  of  a  division 
fence,  will  not  bar  title,  where  it  appears  that  such 
occupation  was  under  a  belief  that  flie  fence  was  on 
a  true  line,  and  without  intention  of  claiming  beyond 
the  true  line,  as  described  in  the  deeds. 

Skinker  v.  Haagsma  (Mo.)  12  S.  W.  659. 


336  A    MANUAL    OF    LAND    SURVEYING. 

86.  Lands  are  not  surveyed  lands  by  the   United 
States   until   a    certified   copy   of   the   official   plat   of 
survey  has  been  filed  in  the  local  land  office. 

United  States  v.  Curtner,  38  F.  I. 

87.  One  who  receives  deeds  of  lots,  and  conveys  to 
others,    according    to    an    unacknowledged    plat    of    a 
town,  is  thereby  estopped  from  denying  the  sufficiency 
of  the  dedication  for  want  of  the  acknowledgment. 

Giffen  v.   City  of  Olathe   (Kan.)   24  P.   470. 

88.  Testimony  of  declarations  of  a  grantor,  before 
the  execution  of  a  deed,  tending  to  establish  a  bound- 
ary other  than  that  made  by  the  deed  as  construed 
by  the  court  on  appeal,  is  inadmissible,  as  its  effect 
would  be  to  convey  land  by  parole  in  contravention  of 
the  statute  of  frauds. 

Harris  v.  Oakley,  7  N.  Y.   S.  232. 

89.  Where  a  town  site  was  surveyed  and  laid  out 
in  lots,  blocks,  streets  and  alleys,  and  a  plat  thereof 
made    and   lithographed,    and    distributed    among   thy 
occupants  of  the  town  site,  and  one  of  the  lithographed 
copies   was   afterwards   recorded   in-  the    office    of  the 
register  of  deeds,  but  the  same  was  not  acknowledged, 
and  the  town  site  was  pre-empted  by  the  president  of 
the  town  site  company,  and  a  patent  was  obtained  by 
him  for  the  benefit  of  the  occupants,  under  the  town^ 
site  act  (5  U.  S.  St.  657),  there  was  a  sufficient  dedica- 
tion of  the  streets  and  alleys  of  said  town,  despite  the 
want  of  acknowledgment  of  the  recorded  plat. 

Giffen  v.  City  of  Olathe   (Kan.)   24  P.   470. 

90.  A  deed  conveying  land  in  a  town,  but  "  reserv- 
ing streets  and  alleys  according  to  recorded  plat  of  the 
town,"  passes  the  fee  in  such  streets  when  such  fee  was 
at  the  time  held  by  the  grantor  subject  to  the  ease- 
ment of  the  public  therein. 

Gould  v.  Howe  (111.)   23  N.  E.  602. 


ALLUVIUM.  337 

91.  Where  surveys  of  1837  and  1856  do  not  agree 
the  former  holds. 

Palmer  v.  Montgomery-,  26  X.  Y.  Rep.   536. 

92.  The  boundary  lines  of  water  lots  fronting  on  a 
river  extend  into  the  river  at  right  angles  with  the 
thread  of  the  stream,  without  reference  to  the  shape 
of  the  shore. 

Clark  v.  Campau,  19  Mich.  328. 

Bay  City  Gas  Light  Co.  v.   Ind.  Works,  28  Mich.  182. 

Twogood   v.    Hoyt,   42    Mich.    609. 

Noms  v.  Hill,   1  Mich.  202. 

93.  Where  a  certain  distance  is  called  for  from  a 
given  point  on  a  navigable  stream  to  another  point 
on  the  stream  to  be  ascertained  by  measurement,  such 
measurement  must  be  made  by  its  meanders,  and  not 
in  a  straight  line.      The  same  rule  prevails  when  dis- 
tance is  called  for  along  a  traveled  highway.     A  dif- 
ferent rule  is  sometimes  adopted  when  the  stream  is 
not  navigable.     When  a  tract  of  land  is  bounded  upon 
a  navigable  stream,  the  distance  upon  the  stream  will 
be   ascertained,   in   the   absence   of    other   controlling 
facts,  by  measuring  in  a  straight  line  from  the  oppo- 
site boundaries. 

People  v.  Henderson,  ,40  C'al.  29. 

94.  In  computing  the  number  of  acres  in  a  survey, 
"from,"    "to,"    and   "with"    the   bank   of   a    stream 
mean  to  low-water  mark. 

Lamb  v.  Ricketts,  11  Ohio    311. 

1.  Alluvium  means  an  addition  to  riparian  land 
gradually  and  imperceptibly  made  through  causes 
either  natural  or  artificial  by  the  water  to  which  the 
land  is  contiguous.  It  matters  not  whether  the  addi- 
tion be  on  streams  which  overflow  their  banks,  or  on 
those  which  do  not.  In  each  case  it  is  alluvium. 

County  of  St.  Clair  v.  Livingston,  23  Wall.   (U.   S.)  46. 


338  A  MANUAL  OF  LAND   SURVEYING. 

2.  Land  formed  by  alluvium  in  a  river  is  in  gen- 
eral to  be  divided  among  the  several  riparian  owners 
entitled  to  it,  according  to  the  following  rule:     Meas- 
ure the  whole  extent  of  their  ancient  line  on  the  river, 
and  ascertain  how  many  feet  each  proprietor  owned  on 
this  line.     Divide  the  newly  formed  river  line  into  an 
equal  number  of  parts,  and  appropriate  to  each  owner 
as  many  of  these  parts  as  he  owned  feet  on  the  old 
line;  and  then  draw  lines  from  the  points   at  which 
the  proprietors  respectively  bounded  on  the  old,  to  the 
points  thus  determined  as  points  of  division  on  the 
newly    formed    shore.     This    rule    is    to    be    modified 
under  particular  circumstances;   for   instance,   if  the 
ancient   margin  has  deep  indentations  or  sharp  pro- 
jections, the  general  available  line  of  the  river  ought 
to  be  taken,  and  not  the  actual  length  of  the  margin 
as  thus  changed  by  the  indentations  or  projections. 

Deerfield  v.  Arms,  17  Pick.  Mass.  41. 

Jones,  et.  a!,   v.   Johnston,  18   How.    (U.   S.)    100. 

3.  Alluvium  deposited  against  an  island  in  a  lake 
and  a  neighboring  lot,  so  as  to  connect   them,  must 
be  equally  divided  between  the  owners  of  both. 

Bigelow  v.  Hoover  (Iowa)  52  N.  W.  124.. 

4.  Flats  situate  in  a  tidal  river  at  a  point  in  its 
course  above  the  line  of  low  tide,  are  to  be  divided 
among  the  adjoining  properties,  by  drawing  lines  from 
the  terminal  of  the  latter  on  the  banks  at  the  ordinary 
stage  of  water  to  and  at  right  angles  with  the  centre 
line  of  the  river. 

Tappan    v.    Boston    Water    Power    Co.     (Mass.)    31    N.    E.    703; 
Browne  v.  Same  Id. 

5.  Under   Kev.   Stat.  U.   S.   §2396.     Held,  that  in 
surveying  a  lot  bordering  on  a  river  the  water-course 
becomes  the   boundary,   and   continues  so,  no  matter 


ALLUVIUM.  339 

how  much  it  shifts  by  accretion,  and  conveyances  of 
the  lot  pass  all,  including  such  accretion  to  that  line. 

East   Omaha   Land   Co.   v.  Jeffries,   40  F.    386. 

6.  The  facts  that  rapid  changes   in  the   banks'  of 
the  Missouri  River  are  constantly  going  on,  and  that 
40  acres  have  been  added  to   adjoining  land,   do  not 
overthrow  an  averment  of  a  bill  to  quiet  title  to  such 
addition,  on  the  ground  of  accretion,  that  it  was  by  an 
imperceptible  increase,  where  it  was  nearly  20  years 
in  forming. 

East  Omaha  Land  Co.   v.  Jeffries,   40   F.    386. 

7.  The  rule  that  owners  of  land  bounded  by  streams 
are   entitled   to    additions   to    their   land    formed   by 
accretion  is  applicable  to  the  Missouri  river,  notwith- 
standing the  peculiar  character  of  that  stream,  and  of 
the  soil  through  which  it  flows,  whereby  changes  in  its 
banks  are  great  and  rapid. 

Jeffries  v.   East  Omaha   Land  Co.,  10   S.  Ct.   518. 

8.  Where  the  official  plat  of  the  survey  of  govern- 
ment lands  shows  a  river  as  one  boundary  of  a  certain 
lot,  in  accordance  with  Rev.  St.  TT.  S.  §2395,  .et  seq., 
a  subsequent  patent  for  the  lot,  describing  it  by  num- 
ber, and  referring  to  the  plat,  on  which  it  is  marked 
as  containing  a  certain  amount,  and  deeds,  describing 
the  lot  by  number,  pass  all  accretion  10  the  lot  up  to 
their  respective  dates. 

Jeffries  v.   East  Omaha  Land  Co.,  10  S.  Ct.  518. 

5.     Rules  Applicable  to   the    United    States  Surveys. 

— "  All  the  corners  marked  in  the  surveys  returned  by 
the  surveyor-general  shall  ~be  established  as  the  proper 
corners  of  the  sections  or  subdivisions  of  sections 
which  they  were  intended  to  designate." 

"  The  boundary  lines  actually  run  and  marked  in 
the  surveys  returned  by  the  surveyor-general  shall  be 
established  as  the  proper  boundary  lines  of  the  sections 


340  A   MANUAL    OF    LAND    SURVEYING. 

or  subdivisions  for  which  they  were  intended ;  and  the 
length  of  such  lines  as  returned  shall  'be  held  and 
considered  as  the  true  length  thereof." 

The  preceding  quotation  from  section  2396  of  the 
Revised  Statutes  of  the  United  States,  settles  all 
questions  in  regard  to  any  change  in  the  corners,  lines 
or  measures  of  the  government  survey.  They  are 
thereby  made  unchangeable,  the  statute  thus  empha- 
sizing the  common  law,  which  holds  the  same  doctrine 
to  be  true  of  all  original  surveys  after  the  land  has 
been  conveyed  in  accordance  with  them.  Hence,  in 
making  resurveys,  the  surveyor  must  find,  if  possible, 
the  original  corners,  and  make  his  courses  and  dis- 
tances agree  with  those  of  the  United  States  survey. 

The  following  points  have  been  decided  by  the  courts 
with  reference  to  these  surveys: 

RULE  1. — The  original  surveys  by  which  the  govern- 
ment sold  its  land  and  conveyed  it  to  the  purchaser 
establish  the  rights  of  the  parties  as  to  the  bound- 
aries. No  line  which  will  vary  the  rights  thus  ac- 
quired can  afterwards  be  established  without  the  con- 
sent of  all  parties. 

May  v.  Baskins,  12  S.  and  M.   (Miss.)   428. 

2.  AH  disputes  as  to  the  boundaries  of  land  are  to 
be  governed  by  the  United  States  surveys,  unless  there 
is  some  statute  to  the  contrary. 

Taylor   v.    Fomby    (Ala.    1897)    22    So.    910. 

3.  Government  corners,  fixed  by   a   United  States 
surveyor,   will   control   the   field   notes   of  the   survey 
taken  at  the  time  the  corners  were  erected,  and  also 
the  field  notes  of  any  subsequent  survey. 

In  the  absence  of  a  government  corner,  or  of 
satisfactory  proof  of  its  location,  the  field  notes  of  a 
government  survey  will  govern,  and  are  prima  facie 
evidence  of  the  true  location  of  the  true  line  of  the 
survey. 

Knoll  v.  Randolph,  92  N.  W.  195   (Neb.  1902). 


SPECIAL     RULES.  341 

4.  Land  sold  under  the  United  States  surveys  pass 
according  to  the  description  of  the  legal  subdivisions, 
whether  those  subdivisions  contain  the  legal  quantity 
or  not,  more  or  less. 

Fulton  v.   Doe,  6  Miss.  751. 

.    -  ':-+  • 

5.  Each   section  or   a  subdivision  of  a   section   is 
independent  of  any  other  section  in  the  township  and 
must    be    governed    by    its    marked    and    established 
boundaries.     Should  they  be  obliterated,  a  last  recourse 
•must  be  had  to  the  best  evidence  that  can  be  obtained 
showing  their  former  situation  and  place. 

Lewen  v.    Smith,   7   Port    (Ala.)    428. 

6.  Field    notes   must    yield   to    actual   monuments 
erected  by  the  original  surveyor.     They   are  only  to 
be  relied  on  as  evidence  to  assist  in  finding  the  exact 
situation  of  the  monuments. 

McClintock  v    Rogers,  11  111.   279. 

7.  The  rule   that  monuments  control  courses  and 
distances  applies  to  discrepancies  in  government  sur- 
veys between  the  courses  and  distances  and  the  witness 
trees  called  for  in  the  field  notes. 

England  v.  Vandermark  (111.  Sup.)  35  N.  E.. 

8.  Monuments    found    at   the   two    extremes    of    a 
township  line  are  entitled  to  no  more  controlling  in- 
fluence in  determining  the  actual  location  of  an  inter- 
mediate line  than  the  section  corners  established  along 
the  line.     All  original  monuments  established  in  con- 
nection with  the  field  notes  and  plats  must  be  referred 
to  in  order  to  define  the  locality  of  the  line. 

McClintock  v.  Rogers,  11  111.  279. 

9.  The    corners    established    by    the    original    sur- 
veyors of   public   lands   by   authority   of  the   United 
States  are  conclusive  as  to  the  boundaries  of  sections 
and  divisions  thereof;  and  no  error  in  placing  them 


342  A    MANUAL    OF    LAND    SURVEYING. 

can  be  corrected  by  any  survey  made  by  individuals  or 
a  state  surveyor. 

Arnier  v.  Wallace,  28  Miss.  556. 

10.  In   ascertaining  the   lost    corner  of   a   section, 
recourse  must  be  had  to  the  unobliterated  marks  of  the 
original  survey,  the  field  notes  and  plats     and  subse- 
quent surveys  made  under  their  guidance.     If  only  a 
portion  of  one  of  the  boundary  lines  leading  to  the 
lost  corner  on  a   township  line  has  been  obliterated, 
the  remaining  portion  must  be  considered  established 
as  marked,  and  the  corner  must  be  presumed,  in  the 
absence  of  evidence  to  the  contrary,  to  be  at  the  point 
where  the   marked  line  if  continued  would  intersect 
the  township  line.     But  if  the  lost  corner  is  proved 
to  have  been  at  another  point,  the  lost  portion  of  the 
boundary  must  be  ascertained  by  running  a  straight 
line  from   the   point   where   the  marks    disappear   to 
that  corner. 

Billingley  v.    Bates,  30  Ala.  378. 

11.  In   determining  the  line  between  the  quarters 
of  a  section,  the  quarter  post  established  by  the  gov- 
ernment surveyors  must  govern  in  all  cases  where  its 
Iocati6n  can  be  ascertained. 

Vroman  v.   Dewey,  23  Wis.   530. 
Britton  v.  Ferry,  14  Mich.  53. 

12.  In  re-establishing  a  lost  quarter  post  on  a  sec- 
tion line,  any  difference  in  the  length  of  such  line  by 
actual  measure  as  compared  with  that  indicated  by  the 
government    survey    should    be    divided    between    the 
parts    in    proportion    to    their    respective    lengths    as 
shown  by  that  survey. 

Jones  v.  Kimble,  19  Wis.   429. 

13.  Where   a   government   corner   is  lost    or   oblit- 
erated, so  that  resort  must  be  had  to  the  government 
field  notes  for  the  purpose  of  determining  its  location, 


SPECIAL     RULES.  343 

but  these  field  notes  are  inconsistent,  and  can  not  be 
reconciled,  there  is  no  universal  rule  that  certain  ones 
shall  be  preferred  to  the  others,  but,  as  in  a  case 
where  living  witnesses  contradict  each  other,  those 
should  be  accepted  as  correct  which,  under  all  the 
circumstances,  are  most  entitled  to  credit,  and  most 
likely  to  be  in  accordance  with  the  actual  facts. 

A  witness  or  bearing  tree  is  not  an  established 
corner,  but  merely  a  designated  object  from  which  in 
connection  with  the  field  notes,  the  location  of  the 
corner  may  be  ascertained. 

Stadin  v^  Helin  (Minn.  1899)  79  N.  W.  587. 

14.  The   unvarying  rule   to   be   followed   in   estab- 
lishing a  lost  corner,  is'  to  start  at  the  nearest  known 
point  on  one  side  of  the  lost  corner,  on  the  line  on 
which  it  was  originally  established;  to  then  measure 
to  the  nearest  known  corner  on  the  other  side,  on  the 
same  line;  then,  if  the  length  of  the  line  is  in  excess 
of  that  called  for  by  the  original  survey,  to  divide  it 
between  the  tracts  connecting  such  two  known  points, 
in  proportion  to  the  length  of  the  boundaries  of  such 
tracts  on  such  line,  as  given  in  such  survey. 

Lewis  v.   Prien   (Wis.   1897)   73  N.  W.  654. 

15.  Where  the  original  survey  and  field-notes  of  a 
township  show  all  the  sections  full,  but,  after  all  the 
natural  monuments  in  the  two  northern  tiers  of  sec- 
tions have  been  lost,  it  appears  that  there  is  a  short- 
age somewhere  within  those  two  tiers,  such  shortage 
will  be  apportioned  between  the  two  tiers,  and  not  im- 
posed wholly  on.  the  northern  tier,  though  the  survey 
was  made  by  beginning  «t  the  southeast  corner  of  the 
township,  and  working  north. 

James  v.   Drew    (Miss.)    9   So.   293. 

16.  If  the  distance  between  recognized  government 
corners  as   originally    established  .  overruns  or  under- 
runs  that  given  in  the  field  notes,  it  should  be  divided 


344  A    MANUAL    OF    LAND    SURVEYING. 

pro  rata  between  the  intervening  sections.  The  origi- 
nal field  notes  should  be  the  main  guide.  Section 
lines  being  frequently  deflected,  the  true  corners  must 
be  tested  by  east  and  west  distances  from  the  recog- 
nized government  corners  yet  standing  in  the  same 
township  as  well  as  by  north  and  south  distances. 

Martz  v.   Williams,  67   111.  306. 

17.  Unknown   corners  must   be  found  by  the   cor- 
roborative   testimony   of    all    known   corners   with    as 
little   departure  as  may  be  from  the  system  adopted 
on  the  original  survey,  without  giving  preponderance 
to  the  testimony  of  any  one  monument  above  ^another. 

In  re-establishing  lost  corners  between  remote  cor- 
ners of  the  same  survey,  when  the  whole  length  of  the 
line  is  found  to  vary  from  the  length  called  for;  we 
are  not  permitted  to  presume  that  the  variance  arose 
from  the  defective  survey  of  any  part,  but  must  con- 
clude in  the  absence  of  circumstances  showing  the 
contrary  that  it  arose  from  the  imperfect  measurement 
of  the  whole  line,  and  distribute  such  variance  be- 
tween the  several  subdivisions  of  the  whole  line  in  pro- 
portion to  their  respective  lengths. 

Moreland  v.   Page,  2   Clarkes,  Iowa,  139. 

18.  Quarter  posts  of  the  government  survey  are  to 
be  as  much  respected  as  the  corners  of  townships  or 
sections  however  distant  from  the  center  line. 

Campbell  v.  Clark,  8  Mo.  558. 

19.  There  was  a  mistake  in  the  government  survey 
of  a  section  by  which  the  quarter  section  line  and  the 
meander  line  of  a  river  were  shown  on  the  official  plat 
to  be  one  and  the  same  line,  being  the  boundary  line 
of  the  fractional  lots.     As  a  matter  of  fact  they  were 
a  considerable  distance  apart.     There  was  no  question 
as  to  the  location  of  the  quarter  section  corners.     In 
a  suit  to  determine  the  ownership  of  the  land  between 


SPECIAL    RULES."  845 

the  quarter  section  line  and  the  river,  it  was  held 
that  'the  quarter  section  line  should  be  adhered  to  as 
the  more  certain  call,  and  that  where  the  lines  of  a 
survey  can  be  run  from  well  ascertained  and  estab- 
lished monuments,  they  are  to  control  and  govern  a 
description  delineated  on  a  plat,  although  the  quantity 
in  the  fraction  fell  short  of  the  amount  laid  down 
in  the  plat  about  as  much  as  there,  was  land  contained 
between  the  quarter  line  and  the  river. 

Martin  v.  Carlin,  19  Wis.  454. 

20.  When  a  deed  designates  the  land  conveyed  as 
one  of  the  subdivisions  known  in  the  United  States 
survey,   as,    for   instance,    a    quarter,   half -quarter   or 
quarter-quarter  section,   the   presumption   is  that   the 
parties  intend  that  the  tract  shall  be  ascertained  in 
the  same  manner  as  is  done  in  the  government  sur- 
veys.    Not  so,  where  the  deed  conveys  a  tract  of  land 
not  known  in  that  system  of  surveys,  as,  for  instance, 
the  east  half  of  a  lot,  or  of  a  quarter-quarter  section. 

Cogan  v.  Cook,  22  Minn.  142.         «- 

21.  The  line  between  the  northeast  and  northwest 
quarters  of  a  quarter  section  is  to  be  extended  south 
from  a  point  midway  between  the  northeast  and  north- 
west corners,  rathe'r  than  from  a  point  on  such  line 
1,320  feet  from  one  of  the  corners. 

Packscher  v.    Fuller    (Wash.)    33   P.   875. 

22.  The  defendant  sold   the  north   half  of  a  lot 
which  is  bounded  on  the  west  side  by  the  Au  Gres 
river.     But  the  river  is  not  straight  at  this  point,  and 
the  north  line  of  the  lot  is  longer  than  the  south  line. 

The  bill  demands  the  north  half  of  the  lot,  and  the 
north  half  must  mean  the  north  half  in  quantity  divided 
from  the  remainder  by  an  east  and  west  line. 

Au  Gres  Boom  Co.   v.  Whitney,  26  Mich.  44. 

15.  It  is  a  question  of  fact  to  be  determined  by  all  the 
surrounding  circumstances  whether  the  land  between  the 


346  A    MANUAL    OF    LAND    SURVEYING. 

meander  line  and  the  shore  of  the  lake  or  water  course  is 
included  in  the  survey. 
Shoemaker  v.  Hatch,  13  Nev.  267. 

23.  The  lines  run  to  divide  sections  into  halves  and 
quarters,  if  erroneous,  may  be  corrected,  for  they  are 
subdivided  by  law;  and  if  the  officer  in  running  the  sub- 
division line  makes  a  mistake,  it  can  be  corrected  by  run- 
ning the  line  according  to  law. 

Nolin  v.  Palmer,  21  Ala.  66. 

24.  An  original  township  was  divided  into  sections  "  by 
running  through  the  same,  each  way,  parallel  lines  at  the 
end  of  every  two  miles,  and  making  a  corner  at  the  end 
of  every  mile,"  arid  afterward  a  supplemental  survey  was 
made  under  a  subsequent  statute,  which  directed  that 
these  two  mile  blocks  should  be  subdivided  by  running 
straight  lines  from  the  corners  thus  marked  to  the  oppo- 
site corresponding  corners.    Held,  that  where  the  original 
mile  corners  in  a  certain  block  can  be  clearly  identified, 
the  courses  of  lines  of  subdivision  within  the  block  can- 
not be  determined  by  proof  of  monuments,  blazes,  or 
other  witness  marks  found  in  other  blocks  in  the  town- 
ship. 

Ginn  v.  Brandon,  29  Ohio  St.  656. 

25.  When  a  navigable  stream  intervenes  in  running  the 
lines  of  a  section,  the  surveyor  stops  at  that  point,  and 
does  not  continue  across  the  river.    The  fraction  thus 
made  is  complete,  and  its  contents  can  be  ascertained. 

Therefore,  when  there  is  a  discrepancy  between  the  cor- 
ners of  the  section  as  established  by  the  United  States, 
and  the  lines  as  run  and  marked,  the  latter  do  not  yield 
to  the  former. 

Lewen  v.  Smith,  7  Port.  (Ala.)  428. 

26.  In  government  surveys,  the  line  actually  run  by  the 
government  surveyors  is  the  true  line. 

Goodman  v.  Myrick,  5  Oregon,  65. 

27.  In  a  case  where  the  township  lines  had  been  run 
and  marked  by  the  United  States  survey,  but  the  field 


SPECIAL    KULES.  347 

notes  of  the  subdivision  lines  were  fraudulent  and  re- 
jected by  the  surveyor-general,  because  incorrect,  no 
proper  survey  of  them  having  been  made,  it  was  held 
that  the  line  between  sections  one  and  two  must  be  ascer- 
tained by  running  a  straight  line  from  the  corner  of  the 
sections  established  on  the  exterior  line  of  the  township 
to  the  corresponding  corner  on  the  opposite  side  of  the 
township. 

Hamil  v.  Carr,  21  Ohio  St.  258. 

28.  Where  the  initial  point  in  the  description  of  prem- 
ises in  a  deed  is  the  southeast  corner  of  the  north  half  of 
the  southeast  quarter,  fractional,  of  a  section,  and  the 
quarter-section  is  made  fractional  by  a  meandered  lake 
so  situated  as  to  cover  the  eastern  and  central  portions 
thereof;  and  the  parcel  described  was  carved  out  of  the 
north  half  within  a  year  after  the  same  was  patented, 
the  southeast  corner  in  question  is  construed  to  be  the 
point  which  constituted  the  southeast  corner  of  the  land 
as  it  was  surveyed  out  and  platted  by  the  government, 
which  located  it  on  the  meandered  line  of  the  lake.    The 
fact  that  the  waters  of  the  lake  have  since  receded  can- 
not change  the  boundaries  as  previously  located. 

Verplanck  v.  Hall,  27  Mich,  79. 

29.  Extending  fractional  lots  beyond  quarter  lines: 
Etheridge  and  Stone  were  the  original  settlers,  pre-empt- 
ors,  and  purchasers  of  fractional  section  22.    Etheridge's 
patent  called  for  "the  S.  W.  ^  of  Sec.  22»  containing  92.67 
acres."    Stone's  patent  called  for  "  S.  E.  subdiv.  Qr.  Sec. 
22,  containing  110.50  acres."    These  two  descriptions  were 
in  controversy  in 

Brown's  lessees  v.  Clements,  3d  How.  650. 

In  the  figure  (page 34 8)  the  full  lines  show  the  frac- 
tional section  as  it  was  returned  on  the  official  plat.  The 
dotted  lines  show  the  quarter  lines  as  they  would  have 
been  if  the  section  had  been  f  ulL 


348 


A    MANUAL    OF    LAND    SURVEYING. 


FIG.  71 


On  the  part  of  the  grantees  of  Etheridge  two  claims 

were  set  up.  One  was  that 
under  the  pre-emption  laws 
Etheridge  was  entitled  to  a 
full  quarter  section  of  land. 
The  other  was  that,  as  his 
deed  called  for  the  S.  W.  % 
and  the  fractional  section 
was  of  such  size  and  shape 
that  a  regular  southwest 
quarter  could  be  laid  out 
from  it,  he  was  entitled  to 
it,  and  that  the  action  of  the 
Surveyor  General  in  returning  irregular  subdivisions  of 
the  section,  when  he  could  have  made  one  regular  quarter 
section  out  of  it,  was  contrary  to  law,  and  therefore  void. 
The  Supreme  Court  by  a  bare  majority  upheld  these 
claims  and  decided  the  case  on  those  grounds. 

The  case  of  Brown's  lessees  v.  Clements  was  decided  in 
1845,  several  of  the  judges  strongly  dissenting  from  the 
decision.  In  1858  the  same  tract  of  land  came  in  question 
again. 

Gazzam  v.  Phillips'  lessee  and  others,  20th  Howard  372. 

Speaking  of  the  sales  to  Stone  and  Etheridge,  the  Court 
says: 

"  The  sales  in  each  case  were  made  in  conformity  with 
the  plat  of  the  survey  then  on  file  in  his  office,"  etc. 

"  We  deny  altogether  the  right  of  the  court  in  this  ac- 
tion to  go  beyond  these  terms  thus  explicit  and  specific 
and  under  a  supposed  equity  in  favor  of  Etheridge, 
arising  out  of  the  pre-emption  laws,  to  the  whole  of  the 
southwest  quarter— enlarge  the  description  in  the  grant, 
or  more  accurately  speaking,  determine  the  tract  and 
quantity  of  the  land  granted  by  this  supposed  equity 
instead  of  by  the  description  of  the  patent. 

"  We  are  not  satisfied  that  there  was  any  want  of  power 
in  the  surveyor  general  in  making  subdivisions  of  this 


SPECIAL    RULES.  349 

section  according  to  the  plat  and  in  conformity  with 
which  the  sales  of  the  lands  in  dispute  were  made. 

"The  Act  of  1820  provides  that  fractional  sections 
containing  160  acres  and  upwards  shall  in  like  manner, 
as  nearly  as  practicable,  be  subdivided  into  half  quarter 
sections  under  such  rules  and  regulations  as  may  be 
prescribed  by  the  secretary  of  the  treasury. 

"The  secretary  of  the  treasury,  on  the  10th  of  June 
following  the  passage  of  the  act,  issued  regulations 
through  the  commissioner  of  the  land  office,  directing 
fractional  sections  containing  more  than  160  acres  to  be 
divided  by  north  and  south  or  east  and  west  lines,  so  as 
to  preserve  the  most  compact  and  convenient  form.  This 
section  was  divided  by  a  north  and  south  line  according 
to  these  instructions.  The  question  came  before  the 
secretary  of  the  treasury  and  before  us  in  1837,  and  the 
construction  first  given  and  the  practice  of  the  surveyor 
general  under  it  confirmed.  Attorney  General  Butler  in 
a  well  considered  opinion  observed:  'If  congress  had 
intended  that  fractional  sections  should  at  all  events  be 
divided  into  half  quarter  sections  when  their  shape  per- 
mitted the  formation  of  such  a  subdivision,  I  think  they 
would  have  said  so  in  explicit  terms,  and  that  the  discre- 
tionary power  entrusted  to  the  secretary  would  have  been 
plainly  confined  to  the  residuary  parts  of  the  section. 
And  further  that  the  clause  in  the  first  section  of  the  act 
of  1820,  concerning  fractional  sections  containing  less 
than  160  acres  (which  are  not  to  be  divided  at  all)  is 
decisive  to  show  that  congress  *  *  did  not  deem  it 
indispensable  that  regular  half  quarter  sections  should  in 
all  practicable  cases  be  formed  by  the  surveyors.  On 
the  contrary,  it  shows  that  they  preferred  a  single  tract 
though  containing  more  than  80  acres  to  small  incon- 
venient fractions.'" 

The  court  adds:  "We  entirely  concur  in  this  construc- 
tion of  the  act,"  and  further  goes  on  to  say:  "The  only 
difficulty  we  have  had  in  this  case  arises  from  the  cir- 
cumstance that  a  different  opinion  was  expressed  oy  a 


350  A    MANUAL    OF    LAND    SURVEYING. 

majority  of  this  court  in  the  case  of  Brown's  lessees  v. 
Clement,  3  How.  650. 

"  It  is  possible  some  rights  may  be  disturbed  by  refusing 
to  follow  the  opinion  expressed  in  that  case,  but  we  are 
satisfied  that  far  less  inconvenience  will  result  from  this 
dissent  than  by  adhering  to  a  principle  which  we  think 
unsound  and  which  in  its  practical  operation  will  unsettle 
the  surveys  and  subdivisions  of  fractional  sections  of  the 
public  land  running  through  a  period  of  some  38  years. 
We  cannot  adopt  that  decision  or  apply  its  principles  in 
rendering  the  judgment  in  this  case." 

30.  Quarter  posts  on  section  lines  where  there  are  double 
sets  of  section  corners :  "  Quarter  section  corners  are  not 
required  to  be  established  on  the  west  boundary  of  the 
western  tier  of  sections  in  a  township,  nor  on  the-  north 
boundary  of  the  north  tier  of  sections  in  a  township  south 
of  and  bordering  on  a  standard  parallel.    The  resurvey 
of  township,  standard,  or  base  lines,  by  the  deputy  sur- 
veyor for  the  purpose  of  establishing  such  quarter-posts, 
is  unnecessary  and  will  not  be  paid  for." 

Instructions  to  surveyors-general  by  Commissioner  Edmunds,  p.  9. 

31.  "Range  lines  are  run  north  or  south  from  the  base 
line,  and  corners  for  sections  and  quarter  sections  are 
established  thereon  at  every  mile  and  half  mile  for  the 
sections  and  quarter  sections  on  the  west  side  of  the  line, 
but  not  for  those  on  the  east  side"    On  township  lines 
"the  corners  of  sections  and  quarter  sections  are  estab- 
lished at  every  80  and  40  chains  for  the  sections  and 
quarter  sections  on  the  north  side  of  the  line,  but  not  for 
those  on  the  south  side" 

Instructions  to  Deputy  Surveyors  of  the  United  States  for  the 
district  of  Illinois  and  Missouri,  1856,  p.  50. 

6.  Decisions  of  the  General  Land  Office 
With  reference  to  Mineral  Surveys.— Plats  and 
field  notes:  Of  surveys  of  mining  claims,  required  to 
disclose  all  conflicts  with  prior  surveys,  giving  areas  of 
all  conflicts. 


DESCRIPTIONS    IX     DEEDS.  351 

In  future,  surveyor-general  will  use  no  coloring  on 
plats. 

Com'r.  (N.)    Nov.  16, 1882.    Circular. 

Location  (of  mine) :  Must  be  marked  on  the  ground  so 
that  its  boundaries  can  be  readily  traced. 

K.  Noonday  M'g  Co.  v.  Orient  M'g  Co.,  G  Saw.,  C.  C.,  299 ;  Myers  et  al. 
v.  Spooner  et  a?.,  55  Cal.  R.  257;  Gleason  v.  N.  White  M'g  Co.,  13  Nev. 
R.,  443;  Southern  Cross  G.  and  S.  M'g  Co.  v.  Europa  M'g  Co.,  15  id.,  383. 

Surface  line :  Agreement  by  adjoining  claimants,  fixing 
surface  boundary  line  between  them,  must  be  construed 
as  extending  such  line  downward,  through  the  dips  of 
the  vein  or  lode,  to  the  earth's  centre. 

Richmond  M'g  Co.  v.  Eureka  M'g  Co.,  103  S.  C.,  389. 

Bearings  and  distances  must  be  given  in  a  survey,  from 
the  respective  survey  corners  to  the  location  corners,  and 
the  same  must  be  shown  on  the  plat. 

Survey:  Of  a  mining  claim  should  show  location  of 
all  improvements  of  a  municipal  nature,  as  blocks,  alleys, 
etc. 

Sec'y  Dec.  18, 1880,  and  Feb.  3, 1881.    Little  Nettie  Lode. 

7.  Descriptions  in  Deeds.  —  Surveyors  are  fre- 
quently required  to  make  surveys  for  the  purpose  of  fur- 
nishing a  description  of  the  land  to  be  conveyed.  Every 
surveyor  of  experience  is  familiar  with  the  many  diffi- 
culties encountered  in  correctly  locating  boundary  lines, 
caused  by  defective,  false  or  impossible  descriptions  in 
the  deeds.  The  description  is  the  controlling  guide  to  the 
surveyor  in  locating  a  man's  possessions  on  the  ground, 
hence  it  is  important  that  it  should  be  clear,  distinct  and 
harmonious  in  its  terms. 

Where  land  is  conveyed  in  the  regular  subdivisions  of 
the  United  States  survey,  little  difficulty  will  be  met  in 
writing  a  correct  description.  The  main  caution  to  be 
observed  is  to  avoid  the  common  clerical  error  of  using 
the  wrong  letter  or  word,  such  as  north  instead  of  south, 
or  east  instead  of  west,  thereby  locating  the  deed  in  a 
different  place  from  which  it  was  intended.  Scrutinize 


352  A    MANUAL    OF    LAND    SURVEYING. 

the  description  closely  to  see  that  no  such  error  is  made, 
and  write  plainly,  so  that  no  one  need  make  a  mistake 
in  reading  or  copying  the  description.  A  great  many  of 
these  mistakes  are  caused  by  bad  penmanship. 

Similar  remarks  apply  to  the  description  of  land  by 
plat,  where  only  clerical  errors  are  likely  to  be  made. 

It  is  in  the  description  "  by  metes  and  bounds  "  and  by 
courses  and  distances,  that  greatest  care  should  be  taken. 

Do  not  use  two  descriptions  if  one  will  clearly  describe 
the  land.  Avoid  surplusage  and  conflicting  descriptions. 
If  after  writing  a  description  it  is  found  necessary  to 
explain  it,  lay  it  aside  and  if  possible  write  a  description 
that  does  not  need  explanation. 

Let  the  starting  point  be  well  denned  and  permanent, 
so  that  there  need  be  no  difficulty  in  locating  it  at  any 
time  in  the  future.  A  striking  example  of  a  disregard  of 
this  principle  was  brought  to  the  attention  of  the  writer 
when  he  was  called  to  locate  the  boundary  lines  of  several 
lots  in  a  village.  The  descriptions  all  referred  back  to 
a  small  cherry  tree  as  a  starting  point.  The  lines  had 
never  been  marked  on  the  ground  even  by  fences,  and  the 
cherry  tree  had  been  gone  so  long  that  no  one  could  be 
found  who  could  remember  that  there  ever  was  such  a 
tree. 

Not  only  the  starting  point  but  as  many  of  the  angles 
in  the  boundary  as  possible  should  be  described  by  some- 
thing permanent  and  definite  on  the  ground.  This  is 
of  prime  importance.  Let  it  be  the  plainest  and  most 
permanent  that  the  nature  of  the  case  permits. 

If  the  courses  are  given  by  compass  bearings,  state 
whether  they  refer  to  the  magnetic  or  some  other  merid- 
ian. This  is  put  in  the  form  of  a  statement  of  the  decli- 
nation of  the  needle,  written  for  example,  Var.  4°  2(K  E. 
By  this  it  is  understood  that  the  magnetic  meridian 
makes  an  angle  of  4°  20'  to  the  east  of  the  meridian  of 
the  survey.  It  was  formerly  a  custom  to  refer  all  lines  to 
the  magnetic  meridian.  Since  the  adoption  of  the  system 
of  the  United  States  Land  Surveys  it  has  become  a 


DESCRIPTIONS    IX     DEEDS.  353 

custom,  especially  in  that  part  of  the  country  surveyed 
under  that  system,  to  refer  all  surveys  to  the  true  merid- 
ian, or  what  was  supposed  to  be  so.  As  time  has  passed 
and  old. descriptions  have  been  retained  in  the  deeds 
conveying  the  land  from  owner  to  owner,  it  has  become 
impossible  in  thousands  of  cases  to  tell  what  meridian 
controls  the  description.  Hence  we  see  the  prime 
importance  of  permanent  monuments  describing  the 
boundaries,  and  of  describing  the  meridian  of  the  survey. 
If  we  must  needs  figure  out  courses  from  the  change  in 
direction  of  the  needle,  let  us  have  something  definite  to 
start  from. 

Do  not  describe  a  boundary  solely  by  reference  to  the 
boundary  of  the  adjoining  tract,  if  it  can  be  avoided 
without  error.  Such  a  description  requires  the  finding  of 
the  description  of  the  adjoining  tract  whenever  a  survey 
is  made,  and  may  cause  great  delay  and  trouble  before 
the  correct  definite  description  can  be  found.  The  writer 
knows  of  a  case  where  the  only  description  of  the  bound- 
ary line  between  two  village  lots  in  either  deed  is  by  a 
reference  to  the  other:  A.'s  land  is  bounded  on  the  east 
by  B.'s  land,  and  B.'s  land  is  bounded  on  the  west  by  A.'s 
land— nothing  more. 

If  a  boundary  line  is  not  intended  to  be  a  straight  line, 
but  to  follow  a  fence,  a  wall,  a  hedge  or  a  stream,  say 
so  in  the  description.  Hake  everything  clear,  definite, 
concise  and  consistent  throughout,  so  that  a  surveyor 
having  the  description  in  the  deed  can  locate  the  boun- 
daries on  the  ground,  without  having  to  hunt  up  descrip- 
tions from  other  deeds. 

8.  Illustrations.— 1.  "  The  east  half  of  the  northeast 
quarter  of  Section  16,  Township  2  south,  Range  10  west? 

The  United  States  land  department  in  selling  land  in 
regular  subdivisions  of  non-fractional  sections  does  not 
state  the  quantity  in  the  patent.  It  is  quite  customary 
in  later  conveyances  to  add  something  like  the  following: 
"containing  80  acres,  more  or  less,  according  to  the 
United  States  survey."  ^Nothing  is  gained  by  the  addi- 

23 


354  A    MANUAL    OF    LAND    SURVEYING. 

tion.  There  is  a  good  deal  of  useless  verbiage  and  repe- 
tition in  deeds,  the  only  effect  of  which  is  to  add  to  the 
expense  of  making  out  and  recording  them. 

2.  "  The  north  fractional  half  of  the  northeast  frac- 
tional quarter  of  Section  3,  Township  3  south,  Range  9 
west,  containing  98.72  acres,  according  to  the  official  plat 
of  the  United  States  Survey." 

The  area  of  fractional  lots  is  stated  in  the  United 
States  patents.  The  word  fractional  is  used  and  the 
area  given  to  show  that  the  land  is  conveyed  according 
to  the  system  of  the  United  States  survey.  Without 
them  the  description  would  convey  the  aliquot  part  of 
the  entire  area  of  the  section  in  the  same  manner  as 
Description  No.  1. 

3.  "The  south  fraction  of  the  southeast  quarter  of 
Section  28,  Township  ft  north,  Range  3  west,  containing 
in. 85  acres" 

Sections  are  made  fractional  by  streams,  lakes  and 
reservations,  making  fractional  lots  of  all  manner  of 
sizes  and  shapes.  The  land  department  attaches  small 
outlying  fractions  to  the  adjacent  larger  ones,  and  sells 
the  whole  under  one  description,  which  takes  its  name 
from  the  larger  lot.  The  above  description  might  con- 
tain land  attached  from  the  southwest  quarter.  Such 
descriptions  do  sometimes  contain  land  attached  from 
other  sections,  and  even  from  other  townships.  The 
official  plat  of  the  section  shows  precisely  what  land  is 
included  in  the  description. 

4.  "A  piece  of  land  twenty  feet  wide  off  from  the  ewtt 
side  of  Lot  99  of  the  lithographed  plat  of  the  milage  of 
Kalamazoo" 

A  description  like  the  above  sometimes  leads  to  contro- 
versy. Suppose  the  original  survey  by  which  the  lots 
were  laid  out,  was  made  with  a  long  chain,  as  it  was  in 
Kalamazoo,  and  that  there  was  a  surplus  in  the  lot.  The 
purchaser  might  claim  that  he  was  entitled  under  the 
common  law  to  his  proportional  share  of  the  surplus, 
while  the  seller,  if  he  owned  the  balance  of  the  lot, 
might  claim  it  all  as  his  own.  Such  questions  do  fre- 


DESCRIPTIONS    IN     DEEDS.  355 

quently  arise,  and  it  is  better  to  settle  them  at  the  outset, 
by  putting  it  definitely  in  the  description  what  is  meant. 
In  the  above  case  suppose  the  recorded  width  of  the  lot 
to  be  sixty  feet;  then  a  description  calling  for  the  "east 
one-third  of  Lot  99  "  would  show  clearly  that  any  surplus 
or  shortage  in  the  lot  was  to  be  divided,  while  a  descrip- 
tion reading  "  20  feet  off  the  east  side  of  Lot  99,  etc.,  as 
surveyed  by  F.  H.,  May  22nd,  1883,"  would  show  that  the 
later  surveyor's  measure  was  to  govern.  The  care  and 
accuracy  of  measurement  of  land  in  cities  keeps  pace 
with  its  increase  in  value,  and  as  a  careful,  accurate 
measure  cannot  be  expected  to  agree  with  a  careless, 
inaccurate  one,  it  is  best  to  settle  such  questions  in 
advance,  as  far  as  possible. 

5.  "Commencing  at  a  stone  with  a  hole  drilled  in  it,  set 
in  the  east  and  west  quarter  line  of  Section  18,  Township 
4  south.  Range  10  west,  22  chains  east  of  the  range  line, 
from  which  stone  a 

White  oak  16  inches  diameter,  bears  S.  28°  W.t  62  links 
distant,  and  running  thence  (Far.  2°  40*  E.,  at  10  A.  M., 
June  12th,  1880\  north  22°  east  12.00  chains  to  a  stone 
marked  with  a  ci'oss,  set  in  an  angle  of  a  hedge; 

Thence  east  along  the  hedge  8.00  chains  to  an  iron  stake 
of  1%  inch  gas  pipe,  driven  on  west  bank  of  a  ditch; 

Thence  south  along  the  bank  of  the  ditch  5.00  chains  to 
an  iron  stake  of  gas  pipe  driven  in  the  bank  where  the 
ditch  turns  east; 

Thence  south  22°  west  6.61  chains  to  a  stake  set  in  the 
quarter  line,  from  which  a 

Burr  Oak  12  in.  di.  bears  N.  16°  E.,  26  Iks.  distant, 

Burr  Oak  18  in.  di.  bears  8. 46°  E.,  51  Iks.  distant; 

Thence  west  along  the  quarter  line  1031  chains  to  the 
place  of  beginning." 

This  is  given  as  a  sample  of  a  description  by  metes  and 
bounds  such  as  a  surveyor  may  furnish  under  the  ordinary 
circumstances  when  called  on  to  make  a  survey  for  that 
purpose,  and  such  as  he  or  any  other  surveyor  would  have 
no  trouble  in  locating  on  the  ground  at  any  future  time 
so  long  as  any  of  the  monuments  or  bearing  trees  could 
be  found. 


356  A    MANUAL,    OF    LAND    SURVEYING. 


CHAPTEE  XL 

BE-LOCATION   OF   LOST   CORNERS. 

The  general  principles  to  be  observed  in  re-locat- 
ing lost  corners  are  laid  down  in  the  Supreme  Court  deci- 
sions which  have  already  been  quoted. 

A  corner  is  not  lost  so  long  as  its  position  can  be  deter- 
mined by  evidence  of  any  kind  without  resorting  to  sur- 
veys from  distant  corners  of  the  same  or  other  surveys. 
Often  after  making  a  survey  from  a  distant  corner,  the 
surveyor  will  come  upon  some  traces  or  evidence  which 
will  enable  him  to  determine  the  true  position  of  the 
corner  he  is  seeking.  It  is  an  uncertain  way  at  the  best 
to  locate  corners  by  running  lines  and  measuring  from 
distant  corners,  and  should  only  be  resorted  to  in  absence 
of  better  proof  of  the  original  location  of  the  corner 
sought. 

It  will  sometimes  happen  that  the  exact  spot  where  a 
lost  corner  stood  cannot  be  found  or  shown  by  evidence, 
but  it  can  be  proved  that  it  stood  within  certain  limits. 
In  these  cases,  which  are  not  rare,  there  is  no  question 
but  that  the  corner  should  be  placed  at  that  point  within 
the  known  limits  which  best  agrees  with  all  the  evidence 
in  the  case. 

"Failing  of  better  evidence  by  which  to  determine  the 
location  of  a  lost  corner,  we  may  next  resort  to  the  fol- 
lowing methods: 

.  GENERAL  RULE. — Retrace  the  known  lines  of  the  de- 
scription and  find  how  the  lengths  and  directions  of  these 
lines  by  your  survey  agree  with  those  of  the  same  lines 
as  laid  down  in  the  original  description.  Then  run  the 


RELOCATION    OF    LOST    CORNERS.  357 

unknown  lines  and  place  the  lost  corners  so  that  they 
will  bear  the  same  relation  to  the  known  lines  and  cor- 
ners as  they  are  required  to  do  by  the  description  of  the 
original  survey. 

Example.— The  four  lines  of  a  description  are  as  fol- 
lows: 

1.  North   7°  east  12.00  chains. 

2.  South  83°  east    6.00      "  . 

3.  South    7°  west  12.00      " 

4.  North  83°  west    6.00      " 

The  first  line  and  its  termini  are  known.  We  retrace 
that  line  and  find  by  our  survey  that  it  runs  north  7°  30' 
east  and  12.24  chains. 

We  would  then  run  the  remaining  lines,  making  them 
as  follows: 

2.  South  82°  30'  east    6.12  chains. 

3.  South    7°  30'  west  12.24    i*ir 

4.  North  82°  30'  west   6.12   ;  *•  < 

Or  the  compass  may  be  set  on  the  known  line  and  the 
vernier  so  adjusted  that  the  reading  of  the  needle  shall 
be  the  same  as  that  given  in  the  original  description  and 
the  remaining  lines  run  accordingly. 

2.  Be-looation  of  Lost  Corners  of  the  United 
States  Survey. 

RULE  1. — On  base  lines,  correction  parallels,  township 
and  range  lines.  Restore  the  lost  corner  in  line  between 
the  nearest  known  corners  on  the  same  line  and  at  dis- 
tances from  them  proportional  to  those  laid  down  in  the 
field  notes  of  the  government  survey. 

This  rule  supposes  the  original  line  to  have  been  a 
straight  line.  As  a  matter  of  fact  this  is  frequently  not 
the  case.  If  there  is  reason  to  suspect  the  line  to  have 
angles  in  its  course,  measures  from  known  corners  to  the 
right  and  left  of  the  line  will  aid  in  determining  its  true 
position. 

RULE  2. — Lost  closing  section  corners  upon  a  town- 
ship or  range  line,  where  the  closing  distance  from  the 


358  A    MANUAL    OF    LAKD    SURVEYING. 

adjacent  corners  is  not  given  in  the  field  notes  should  be 
restored  by  prolonging  the  known  portion  of  the  line  to 
its  intersection  with  the  township. or  range  line. 

RULE  3.  Lost  interior  section  corners  should  be 
restored  at  distances  from  the  nearest  known  corners, 
north,  south,  east  and  west,  proportional  to  those  laid 
down  in  the  field  notes  of  the  original  survey. 

This  rule  supposes  that^the  measurements  of  the  origi- 
nal survey  were  uniform  on  the  several  adjacent  sections. 
This  is  frequently  not  the  case,  and  it  will  be  well  for  the 
surveyor  to  compare  his  chaining  on  each  section  with 
the  original  measure  between  known  corners  of  the  same 
sections,  choosing  by  preference  those  lines  which  on  the 
government  survey  were  measured  next  previous  to  the 
portion  of  the  line  closing  on  the  lost  corner. 

RULE  4.— Lost  township  corners,  when  common  to 
four  townships,  are  to  be  restored  in  a  similar  manner  to 
interior  section  corners,  Rule  3.  When  common  to  only 
two  townships,  they  are  to  be  restored  according  to  Rule  1. 

RULE  5.— Lost  quarter  section  corners  are  to  be  re- 
stored in  line  between  the  section  corners  which  stand  on 
the  same  line  and  at  distances  between  them  proportional 
to  those  returned  in  the  field  notes  of  the  government 
survey. 

RULE  6. — Lost  meander  corners  are  to  be  restored  by 
running  the  line  from  the  nearest  known  corner  the  di- 
rection and  distance  called  for  by  the  notes  of  the  orig- 
inal survey.  When  a  portion  of  the  line  leading  to  the 
meander  corner  is  known,  it  should  be  prolonged  in  the 
same  direction.  When  no  portion  of  the  line  is  known' 
the  surveyor  will  have  to  use  his  own  judgment  as  to 
what  method  under  the  circumstances  of  the  case  will 
most  nearly  retrace  the  original  line  to  the  corner. 

There  is  no  rule  which  will  rigidly  and  inflexibly  apply 
to  all  cases  for  restoring  lost  corners  and  boundary  lines 
except  this — that  the  aim  of  the  surveyor  should  always 


RELOCATION  OF  LOST  CORNERS.        359 

be  to  find  the  exact  spot  where  the  original  corner  or  line 
was  located.  The  thing  to  find  out  is  not  where  the  cor- 
ner or  line  ought  to  have  been,  but  where  it  actually  was. 

There  are  many  cases  in  which  other  methods  for  re- 
storing any  of  the  corners  mentioned  will  prove  more 
satisfactory  than  the  rules  heretofore  given. 

For  instance,  a  half-quarter  post  properly  planted  at  a 
time  when  both  the  section  and  quarter-section  corners 
adjacent  were  known,  may  be  used  in  restoring  either  of 
these  corners  when  lost,  by  prolonging  the  line  over  the 
known  corners  and  doubling  the  distance.  Any  other 
intermediate  corner  whose  location  is  definitely  known 
may  be  used  in  a  similar  manner.  On  a  similar  principle, 
the  Supreme  Court  of  Illinois  decided  in  the  case  of  Noble 
0,  Chrisman  (88  111.  186)  that  the  northwest  corner  of  sec- 
tion 19  could,  in  that  instance,  be  better  determined  by 
tracing  the  section  lines  from  known  corners  east  and 
west  of  the  range  line  to  their  intersection  with  that 
line,  and  measuring  the  jog  between  the  corners,  than  it 
could  by  prorating  six  miles  of  the  range  line. 

Most  of  the  difficulties  which  the  surveyor  has  to  con- 
tend with  in  restoring  lost  corners  arise  from  errors  made 
in  the  original  survey,  or  in  the  field  notes  thereof.  He 
should  bear  in  mind  that  errors  in  the  original  survey 
cannot  be  corrected  by  him.  In  any  case  of  a  lost  corner, 
find  as  many  of  the  adjacent  corners  of  the  original  sur- 
vey as  possible,  according  to  the  .best  evidence  that  can 
be  had  to  prove  their  exact  location.  Having  done  this, 
the  others  may  be  found  according  to  the  rules  already 
laid  down.  But  do  not  give  up  a  corner  as  lost  while  any 
means  of  finding  its  exact  location  are  left  untried.  There 
is  great  virtue  in  a  pick  and  shovel  intelligently  applied 
to  the  finding  of  corner  posts  and  monuments.  This  is 
very  important,  as  it  is  very  difficult,  if  not  impossible,  in 
many  cases,  to  re-locate  a  lost  corner  in  the  exact  position 
it  originally  occupied,  by  surveys  from  distant  corners. 
The  following  extracts  from  a  paper  read  by  the  author 


360  A    MANUAL    OF    LAND    SURVEYING. 

before  the  Michigan  Association  of  Surveyors  and  Engi- 
neers, treat  more  fully  of  the  application  of  the  foregoing 
principles  to  finding  corners  of  the  United  States  survey 
in  those  regions  where  wooden  posts  were  planted  for 
corner  monuments : 

"  It  often  happens  that  one  surveyor  will  fail  utterly  in  finding  the 
marks  of  an  origina*  corner,  while  another,  more  apt  in  discovering 
the  evidences,  will  strike  upon  it  readily.  These  evidences  are  of -vari- 
ous kinds,  some  of  which  it  is  the  principal  aim  of  this  paper  to  dis- 
cuss. 

I  take  it  that  the  best  possible  evidence  of  the  location  of  an  orig- 
inal corner  is  the  monument  fixed  at  that  corner  when  the  survey  was 
made .  ( Vide  McClintock  v.  Rogers,  11  111.  279 ;  also  Gratz  v.  Hoover,  16 
Penn.  State  Rep.  232 ;  16  Ga.  141.)  After  this  come  witness  trees,  fences, 
distant  corners  of  the  same  survey,  and  the  testimony  of  persons. 

All  these  latter  kinds  of  evidence  only  go  to  corroborate  the  first, 
and  may  take  the  place  of  the  first  only  so  far  as  they  may  any  of  them 
seem  to  have  weight  in  any  particular  case. 

Many  of  the  corners  of  the  United  States  survey  were  marked  by 
planting  a  post  or  stake  in  the  ground.  These  stakes  had  notches  cut 
in  them,  were  squared  at  the  top,  and  set  in  certain  regular  positions 
fn  the  ground.  These  marks  tended  to  distinguish  them  from  other 
stakes  that  might  chance  to  be  driven  in  the  ground  for  any  purpose. 
When  trees  stood  conveniently  near,  two  of  them  were  marked,  and 
their  directions  and  distances  from  the  corner  were  given  in  the  field 
notes.  When  no  trees  were  near,  a  mound  was  sometimes  raised  about 
the  post. 

Some  of  the  posts  have  been  entirely  destroyed,  but  the  bottoms  of 
a  great  many  of  them  still  remain,  much  decayed,  but  plainly  visible 
when  the  surface  earth  is  removed  from  about  them. 

To  find  them,  careful  manipulation  is  required.  The  surveyor  first 
determines  as  nearly  as  he  can,  from  extrinsic  evidence,  the  point 
where  the  corner  post  should  be  looked  for.  He  then,  with  a  shovel, 
spade  or  hoe,  carefully  removes  the  surface  earth,  a  little  at  a  time, 
being  particular  not  to  strike  deep  at  first  into  the  earth  at  the  level  as 
it  was  when  the  stake  was  set.  The  best  and  sometimes  the  sole  evi- 
dence of  a  corner  has  often  been  destroyed  by  an  ignorant  person 
striking  deep  into  the  ground,  expecting  to  find  a  sound  stake,  and 
casting  away  the  decayed  wood  and  filling  up  the  hole  of  a  rotten  one 
without  observing  it.  If  the  surveyor  is  looking  in  the  right  place,  and 
the  earth  has  not  been  previously  removed,  he  will  soon  come  upon  the 
object  of  his  search ;  but  he  must  be  careful  lest  he  mistake  it.  If  the 
soil  is  a  stiff  clay,  packed  hard,  as  in  a  road,  or  covered  with  a  sward, 
he  will  presently  find  a  hole  of  the  size  and  shape  of  the  stake  which 


RELOCATION  OF  LOST  CORNERS.        361 

made  it.  This  hole  will'contain  the  decayed  wood  of  the  stake,  and  a 
marking  pin  may  be  readily  thrust  to  the  bottom.  By  carefully  scrap- 
ing or  cutting  away  the  earth  from  the  top,  or  cutting  down  at  one  side 
of  the  hole,  its  size,  shape  and  direction  may  be  readily  discovered. 
Thus  it  often  happens  that  the  position  of  a  corner  is  as  well  and  sat- 
isfactorily marked  by  the  decayed  stake  as  it  was  by  the  sound  one.  It 
sometimes  happens  that  new  stakes  have  been  driven  beside  the  orig- 
inal stake,  so  that  several  different  ones  will  be  found  by  the  surveyor. 
He  will  seldom  have  any  difficulty  in  deciding  which  is  the  true  corner 
by  its  appearance,  for  the  first  stake  will  be  more  completely  decayed 
and  of  a  darker  color. 

As  a  rule,  it  will  be  driven  deeper  and  straighter  down  than  the 
newer  stakes.  Then,  too,  the  original  stakes  were  generally  round, 
being  cut  from  whole  timber,  while  the  later  ones  were  often  cut  from 
rails  or  other  split  timber,  the  sharp  corners  of  which  can  be  readily 
seen  in  the  holes  made  by  them. 

There  is  thus  in  the  appearance  of  the  stakes  of  the  United  States 
survey  such  peculiarities  and  such  likeness  to  each  other,  even  when 
far  gone  in  decay,  that  the  experienced  surveyor  will  be  impressed 
with  the  appearance  of  truthfulness  pervading  them,  and  will  seldom 
be  deceived.  This  appearance  of  truthfulness  about  a  stake,  which 
to  a  surveyor  is  one  of  the  most  valuable  parts  of  the  testimony  of 
these  silent  witnesses,  is  something  that  courts  and  juries  can  seldom 
take  cognizance  of,  because,  first,  they  speak  in  a  language  that  courts 
and  juries  do  not  understand,  and  secondly,  the  evidence  is  itself  de- 
stroyed by  the  surveyor  in  the  taking,  and  does  not  come  before  court 
or  jury  in  all  its  freshness,  truth  and  purity.  These  decayed  stakes 
may  be  best  observed  in  the  light-colored  subsoil  after  the  black  sur- 
face mould  has  been  removed.  In  sandy  soil,  the  cavity  made  by  the 
stake  is  gradually  filled  by  the  falling  sand  as  the  wood  decays,  but 
rotten  wood  discolors  the  sand  so  that  where  it  has  not  been  disturbed 
the  position,  size  and  shape  of  the  stake  may  be  readily  traced.  In 
the  black  muck  of  our  marshes  and  river  bottoms  it  is  more  difficult 
to  distinguish  the  stake  near  the  surface,  but  as  the  ground  is  soft  and 
wet  the  stakes  were  driven  deep,  and  we  may  sometimes  find  in  the 
wet,  peaty  subsoil  the  bottom  of  the  stake  so  perfectly  preserved  that 
even  the  scratches  made  in  the  wood  by  nicks  in  the  axe  are  plainly  to 
be  seen.  When  the  stakes  are  constantly  wet,  they  do  not  decay. 

Next  we  consider  the  bearing  or  witness  trees.  These  are  marked 
and  their  directions  and  distances  noted,  in  order  to  assist  in  finding 
the  corner  posts  set  on  the  survey.  These  bearing  trees  are  marked 
with  a  blaze  and  a  notch  near  the  ground  on  the  side  facing  the  corner. 
The  measures  were  taken  from  this  notch.  At  this  time  most  of  the 
living  witness  trees  have  grown  to  such  an  extent  that  only  a  scar  re- 
mains in  sight,  to  indicate  the  point  where  the  notch  was  cut.  In  order 


362  A    MANUAL    OF    LAND    SURVEYING. 

to  get  at  the  notch,  the  superincumbent  wood,  which  is  in  some  cases 
a  foot  in  thickness,  will  have  to  be  cut  away.  It  will  not  often  be 
necessary  to  do  this,  as  we  can  come  sufficiently  near  the  correct  point 
to  find  the  stake  without  it.  But  if  the  stake  has  been  destroyed,  or 
there  are  several  stakes  near,  we  shall  need  to  be  exact,  and  measure 
from  the  notch.  If  the  tree  has  been  cut  down,  and  a  sound  stump 
remains,  the  marks  will  be  easily  exposed.  Sometimes  the  mark  is 
gone,  but  a  part  of  the  stump  is  left.  At  others  the  stump  is  gone,  but 
a  dish-like  cavity  remains  in  the  earth  to  show  where  the  tree  once 
stood.  We  can  almost  always  find  under  and  around  these  cavities 
places  where  the  large  roots  have  penetrated  the  subsoil,  and  thus  be 
able  to  locate  within  a  foot  or  so  the  position  of  the  bole  of  the  tree 
when  standing.  In  looking  for  a  corner  post,  we  may  frequently  as- 
sume for  the  time  being  that  a  certain  stump  or  a  cavity  where  a  tree 
had  stood  was  the  stump  of  or  the  place  occupied  by  a  bearing  tree. 
If  we  then  measure  the  required  direction  and  distance,  and  find  a 
stake,  we  may  reasonably  conclude  that  our  assumption  was  correct. 
Such  assumptions  are  frequently  of  great  assistance  in  finding  corners. 
There  may  be,  and  I  know  there  are  cases,  where  the  original  corner 
stakes  have  been  destroyed,  and  can  be  more  nearly  restored  to  their 
original  position  by  measurements  from  old  stump  bottoms  or  holes  in 
the  ground  than  in  any  other  way.  But  bearing  trees,  however  good 
their  condition,  are  by  no  means  infalible  witnesses  as  to  the  location 
of  a  corner.  Mistakes  in  laying  down  their  direction  or  distance,  or 
both,  are  not  rare.  (See  McClintock  v.  Rogers,  11  Ills,  279.)  A  direc- 
tion may  be  given  as  north  instead  of  south,  east  instead  of  west,  or 
vice  versa.  The  limb  may  have  been  wrongly  read  64°  for  56'.  The 
figures  denoting  the  bearing  may  have  been  transposed  in  setting 
down,  as  53  for  35.  So,  too,  the  chain  may  have  been  wrongly  read,  as 
48  for  52,  the  links  having  been  counted  from  the  wrong  end.  Or  they 
may  have  counted  from  the  wrong  tag,  as  48  for  38.  Mistakes  of  the 
nature  of  these  mentioned  are  common,  so  that  in  working  from  a 
bearing  tree  to  find  a  corner,  and  not  finding  the  stake  at  the  place 
indicated  in  the  notes,  it  will  be  well  to  test  all  these  sources  of  error 
before  giving  up  the  search,  for  as  I  have  said  before,  the  post  planted 
at  the  timerof  the  original  survey  is  the  best  evidence  of  the  corner  it  was 
intended  to  indicate. 

I  next  consider  fences  in  their  relations  to  corners.  (Potts  v  Ever- 
hart,  26  Penn.  St.  Rep.,  493.)  Whether  any  particular  fence  may  be 
depended  on  to  indicate  the  true  line  will  depend  on  the  particular  cir- 
cumstances attending  that  case.  In  a  general  and  rough  way,  a  fence 
will  indicate  to  the  surveyor  where  to  begin  looking  for  his  corner. 
But  the  practice  has  been,  and  still  is  common,  for  the  first  settlers  on 
a  section  to  clear  and  fence  beyond  the  line  in  order  to  have  a  clear 
place  on  which  to  set  their  permanent  fence  when  they  get  ready  to 


RELOCATION  OF  LOST  CORNERS.        363 

build  it.  Afterward  they  forget  where  the  line  is  and  set  the  new  fence 
where  the  old  one  stood.  Many  fences,  too,  were  set  without  any  sur- 
vey or  any  accurate  knowledge  where  the  line  was  and  left  there  to 
await  a  convenient  time  to  have  the  line  established.  So,  too,  where 
the  land  has  been  long  settled  and  occupied,  it  is  a  common  custom  for 
adjoining  land  owners  by  consent  to  set  the  fence  on  one  side  of  the 
true  line,  there  to  remain  until  they  are  ready  to  rebuild,  the  one 
party  to  have  the  use  of  the  land  for  that  time  in  consideration  of  clear- 
ing out  and  subduing  the  old  fence  row.  The  original  parties  fre- 
quently sell  out  or  die,  and  the  new  owners  have  no  knowledge  of  the 
agreement  and  suppose  the  fence  to  be  on  the  true  line.  For  these 
reasons,  fences  should  be  looked  on  with  suspicion,  unless  corroborated 
by  other  evidence,  and  the  surveyor  should  enquire  pretty  closely  into 
the  history  of  a  fence  before  placing  any  great  reliance  on  it  to  deter- 
mine the  position  of  a  corner.  It  may  be  the  best  of  evidence,  or  it 
may  be  utterly  worthless. 

It  not  unfrequently  happens  that  there  are  no  trustworthy  marks 
near  a  corner  to  direct  the  surveyor  in  his  search  for  the  post  or  from 
which  to  replace  it  if  it  be  destroyed.  In  these  cases,  he  must  visit  the 
nearest  corners  he  can  find  in  each  direction  (varying  with  the  circum- 
stances whether  it  be  section  corner  or  quarter  post  he  wishes  to  find 
or  restore),  go  through  the  process  of  identification  with  each  of  them, 
and  then  make  his  point  so  that  it  will  bear  the  same  relation  to  these 
corners  as  did  the  original  corner  post.  Many  very  intelligent  gentle- 
men suppose  that  if  the  surveyor  can  but  find  one  of  the  corners  of 
the  original  United  States  survey  he  can  readily  determine  the  position 
of  all  the  rest  from  it.  They  were  never  more  mistaken  in  their  lives. 
The  continual  change  in  the  direction  of  the  magnetic  needle,  the  un- 
certainty as  to  what  its  direction  was  when  any  particular  line  was 
run,  the  difference  in  the  lengths  of  chains,  and  the  difference  in  the 
men  who  use  them,  introduce  so  many  elements  of  uncertainty  into 
the  operation  as  to  render  it  one  of  little  value,  and  not  to  be  resorted 
to  except  in  the  absence  of  trustworthy  evidence  nearer  at  hand. 

If  it  be  a  section  corner  you  desire  to  find  or  replace,  and  have  ad- 
jacent quarter  posts  in  each  direction  to  work  from,  you  will  not  be 
likely  on  the  one  hand  to  fall  more  than  a  rod  or  two  out  of  the  way, 
and  on  the  other  hand  will  not  be  likely  to  come  within  a  foot  or  two 
of  the  right  place.  This  method  will  assist  you  in  seaching  for  the 
original  stake,  and  if  that  be  destroyed,  and  no  better  evidence  pre- 
sents itself,  may  be  used  to  determine  the  point  where  the  corner 
stake  shall  be  placed.  The  chief  difficulty  in  applying  this  method  to 
determine  corners  arises  from  the  f&ct  that  the  measurements  made 
on  the  original  surveys  were  not  uniform  in  length  on  different  sec- 
tions, and  frequently  not  on  different  parts  of  the  same  section.  I  have 
measured  sections  22  and  23  on  a  level  prairie,  along  the  line  of  high- 


364  A    MANUAL    OF    LAND    SURVEYING. 

ways,  where  no  obstacles  of  any  kind  interfered  to  prevent  accurate 
work.  I  took  the  greatest  possible  care  in  the  chaining  to  have  it  as 
accurate  as  chain  work  can  be  done.  On  the  north  Une  of  section  22 
my  chaining  tallied  exactly  with  that  of  the  United  States  survey,  viz., 
79.60.  On  the  north  line  of  section  23,  my  measure  was  80.96,  that  of 
the  United  States  survey,  80.40— a  difference  of  56  links.  Fortunately, 
all  the  corners  of  the  original  survey  on  this  two  miles  of  line  were 
well  preserved,  and  the  distance  between  quarter  post  and  section  cor- 
ners was  uniform  on  the  same  section  in  both  sections.  But  suppose 
that  a  part  of  them  had  been  lost,  and  it  was  required  to  restore  the 
middle  section  corner  (n.  e.  of  22)  from  the  remaining  ones.  Omit  all 
consideration  of  corners,  north  or  south,  and  there  remain  four  differ- 
ent solutions  of  the  problem,  depending  on  which  corners  were  lost 
and  which  preserved.  Of  these  different  solutions,  one  would  place 
the  corner  9^  links,  one  14  links,  one  18M  links,  and  one  28  links,  all 
east  of  the  true  corner.  This  is  not  by  any  means  an  extreme  instance, 
as  I  have  observed  discrepancies  twice  as  great.  It  is  given  simply  to 
show  how  unreliable  is  the  evidence  drawn  from  distant  corners  of  the 
United  States  survey. 

Lastly,  I  shall  consider  the  evidence  of  living  persons.  [Weaver  v. 
Eobinett,  17  Mo.,  459;  Chapman  v,  Twitchell,  37  Maine,  59;  Dagget  v. 
Wiley,  6  Florida,  482:  Lewen  v.  Smith,  7  Port.  (Ala.),  428;  McCoy  v.  Gal- 
loway, 3  Han.  (Ohio),  283;  and  Stover  v.  Freeman,  6  Mass.,  441.]  Con- 
ceding all  men  to  be  equally  honest  in  their  evidence,  there  is  a  vast 
deal  of  difference  among  them  with  regard  to  their  habits  of  observa- 
tion and  their  ability  to  determine  localities.  Some  have  an  exceed- 
ingly acute  sense  of  locality,  if  we  may  so  call  it,  and  can  determine 
very  accurately  the  position  of  any  object  which  they  have  been  accus- 
tomed to  see;  while  others  seem  to  have  little  or  no  capacity  of  that 
sort.  I  have  found  many  men  who  would  describe  accurately  the  sort 
of  monument  used  to  perpetuate  a  corner,  and  who  would  tell  you  that 
they  could  put  their  foot  on  the  very  spot  to  look  for  it;  but  when  the 
trial  came  I  have  found  but  few  of  them  who  could  locate  the  point 
within  several  feet,  unless  they  had  some  object  near  at  hand  to  assist 
the  memory,  and  even  then  they  would  frequently  fail. 

It  may  happen  where  a  corner  post  has  been  destroyed,  that  its  loca- 
tion can  be  more  nearly  determined  by  the  testimony  of  persons  who 
were  familiar  with  it  when  standing  and  can  testify  to  its  relations  to 
other  objects  in  its  vicinity,  than  in  any  other  way.  But  the  surveyor 
in  receiving  this  testimony  should  ascertain  as  far  as  possible  what  are 
the  habits  of  accurate  observation  and  the  memory  of  localities  pos- 
sessed by  th£  person  testifying,  in  order  to  know  how  much  weight  to 
give  his  testimony." 


MISCELLANEOUS.  365 


CHAPTEE  XII. 

MISCELLANEOUS. 

1.  Questions  of  Practice. — Answers  to  most  if  not 
all  questions  which  arise  in  the  surveyor's  practice  will 
be  found  in  the  Supreme  Court  decisions  which  have 
been  quoted.  The  following  questions  which  have  been 
raised  in  several  surveyors'  associations,  are  given  with 
the  answers  adopted  in  each  case,  or  a  reference  to  the 
law  decision  or  principle  which  governs  it. 

1.  An  interior  section  has  its  quarter  posts  out  of  line 
and  not  at  equidistant  points  between  the  section  corners. 
How  shall  the  centre  be  determined  ? 

Ans.  At  the  intersection  of  straight  lines  from  each 
quarter  section  corner  to  its  opposite  corresponding  cor- 
ner. 

See  page  200,  Sec.  100,  Second. 

2.  How  shall  the  quarter  posts  on  the  north  and  west 
lines  of  the  township  which  were  not  established  by  the 
U.  S,  survey  be  located  ? 

Ans.  The  corners  of  half  and  quarter  sections,  not 
marked  on  the  surveys,  shall  be  placed  as  nearly  as  possi- 
ble equidistant  from  those  two  corners  which  stand  on 
the  same  line. 

See  page  200,  Sec.  100,  First. 

Section  6  is  an  exception  to  this  rule. 

See  page  271. 

3.  Posts  for  lines  closing  on  the  north  and  west  boun- 
daries of  townships  are  often  off  the  boundary  line  to  one 
side  or  the  other.    Shall  the  boundary  line  be  deflected  to 
pass  through  these  posts  ? 


366  A    MANUAL    OF    LAND    SURVEYING. 

Ans.  No.  The  posts  serve  to  show  the  position  of  the 
section  line,  but  the  line  itself  stops  at  the  township 
boundary.* 

Mich.  Surv.  Rep.,  1881. 

4.  Are  the  station  or  line  trees  marked  on  the  govern- 
ment surveys  and  returned  in  the  field  notes,  monu- 
ments of  the  lines? 

Ans.  Yes. 

See  page  200,  Sec.  100,  Second. 

Billingsiey  v.  Bates,  30  Ala.  378. 

5.  How  shall  the  east  and  west  quarter  line  of  section 
30  be  located,  there  having  been  no  quarter  post  set  on  the 
east  side  of  the  section  by  the  U.  S.  survey,  because  of  a 
lake? 

Ans.  Locate  the  west  quarter  post  as  directed  in  the 
answer  to  question  2.  Then  run  the  quarter  line  east  on 
a  course  which  is  intermediate  between  the  courses  of  the 
north  and  the  south  lines  of  the  section. 

See  page  268. 

6.  A  closing  corner  on  the  north  or  west  boundary  of 
the  township  is  lost.    The  field  notes  do  not  give  the  dis- 
tance between  the  closing  corner  and  the  adjacent  corner 
on  the  boundary.    How  shall  it  be  restored  ? 

Ans.  Prolong  the  known  portion  of  the  line  to  its  inter- 
section with  the  boundary  and  there  pat  the  corner. 
See  Billingsiey  v.  Bates,  30  Ala.  378;   see  p.  200, 

*  E.  F.  Best,  acting  commisioner  of  the  General  Land  Office,  in 
an  opinion  given  April  16, 1896,  says:  "  These  cases  seem  to  be  ex- 
ceptions to  the  stringent  law  that  the  original  corners  'must stand 
as  the  true  corners  which  they  were  intended  to  represent, whether 
the  corners  be  in  place  or  not.'  The  improper  placing  of  such  a 
corner  results  in  a  change  of  the  true  line  of  the  standard  parallel 
into  a  broken  line,  if  the  erroneous  corner  must  be  held  valid.  It 
is  the  opinion  of  this  office  that  the  true  intent  of  the  law  is  sub- 
served by  holding  that  such  corners  must  show  the  true  locus  of 
the  line  separating  the  sections  but  cannot  alter  the  position  of 
the  township  line;  hence,  that  all  corners  of  small  tracts  adjacent 
thereto  should  be  set  on  the  true  line  originally  run." 
21 


MISCELLANEOUS:  367 

7.  Should  section  lines  running  north  and  south  be  run 
in  a  straight  line  between  known  corners  to  locate  lost 
corners  on  interior  sections  ? 

Ans.  ^N"ot  unless  the  original  lines  were  actually  straight 
lines  between  the  known  points,  which  they  seldom  are. 
See  Moreland  v.  Page,  2  Clarkes,  Iowa,  139. 
Martz  v.  Williams,  67  111.,  306. 

8.  How  shall  the  half-quarter  corner  on  the  quarter  line 
be  located  on  those  quarter  sections  which-  adjoin  the 
north  and  west  lines  of  the  township  ? 

Ans.  Measure  the  distance  from  the  centre  of  the  sec- 
tion to  the  quarter  post  on  the  township  line. 

Then  place  the  corner  on  the  quarter  line  at  a  distance 
of  twenty  chains  proportionate  measurement  from  the 
centre  of  the  section.  In  order  to  prorate  the  distance, 
your  own  measure  should  be  compared  with  a  distance 
which  is  a  mean  between  the  distances  given  in  the  field 
notes  as  the  length  of  the  corresponding  lines  of  the  sec- 
tion on  either  side.  Por  example,  on  section  3  the  dis- 
tance by  U.  S.  survey  from  the  east  £  post  to  township 
line  is  42.18;  from  the  west  J  post  to  township  line  is 
43.20;  which  gives  a  mean  distance  of  42.69. 

Commissioner  McFarland  gives  the  following  reply  to 

a  similar  question* 

DEPARTMENT  OF  THE  INTERIOR,  ) 

GENERAL  LAND  OFFICE, 
Washington,  D.  C.,  February  11, 1882. } 

Isaac  Teller,  Esq.,  Webbervffle,  Ingham  County,  Michigan: 

SIR— I  am  in  receipt  of  your  letter  of  the  5th  instant  requesting  in- 
formation in  regard  to  the  proper  method  of  locating  the  quarter-quar- 
ter corners  north  of  the  legal  centres  of  the  northern  tier  of  sections 
in  a  township  when  the  present  measurement  of  the  east  and  west 
boundaries  of  the  section  differs  from  the  original  measurement, 

In  reply,  I  have  to  state  that  the  length  of  the  quarter  line  from  the 
south  quarter  corner  to  the  township  line  is  to  be  considered  as  the 
mean  of  the  east  and  west  boundaries  of  the  section  as  given  in  the 
field  notes,  and  where  the  present  measurement  of  the  section  lines 
differs  from  the  original  measurement,  the  rule  of  proportionate 
measurement  applies  to  the  quarter  line  as  well  as  to  the  section  lines 
in  the  establishment  of  quarter-quarter  corners  on  the  half  mile  closing 


368 


A    MANUAi,    OF    LAND    SURVEYING. 


on  the  township  boundary.    See  enclosed  circular  dated  November  1, 
1879. 

The  mean  width  of  the  north  half  of  the  section  in  the  case  stated 
by  you  is  40.18  chains,  while  by  your  chaining  it  is  42.42  chains  (calling 
the  distance  to  the  east  and  west  quarter  line  40.00  chains),  therefore 
the  proportion  will  be  as  40.18  : 42.42  : :  20.00  :  21.11  chains,  the  distance 
north  of  the  centre  of  the  section  at  which  by  your  chaining  the  quar- 
cer-quarter  corner  should  be  located. 

Very  respectfully, 

N.  C.  McFAELAND,  Commissioner. 

9^  In  surveying  sections  fractional  on  the  township  line 
to  restore  lost  quarter  section  corners,  should  the  lines  be 
divided  pro  rata  according  to  the  U.  S.  field  notes,  or 
should  the  south  or  east  quarters  be  made  full  and  the 
entire  excess  or  deficiency  be  thrown  into  the  fraction  ? 

Ans.  Any  difference  between  your  measure  and  the 
government  measure  must  be  distributed  proportionally 
between  the  different  parts  of  the  section. 

See  p.  200,  Sec.  100,  Second. 

Moreland  v.  Page,  2  Clarkes,  Iowa  139. 

Jones  v.  Kimble,  19  Wis.  429. 

Martz  v.  Williams,  67  111.  306, 

In  Missouri,  the  Supreme  Court  holds  (Knight  v.  Elliott,  67  Mo.  317) 
a  different  view,  viz.,  that  the  difference  in  measure  is  all  to  be  thrown 
into  the  fraction. 

It  is  difficult  to  see  upon  what  grounds  this  decision  can  be  upheld 
in  view  of  the  fact  that  all  rights  to  the  land  were  acquired  and  held 
under  the  law  of  Congress,  which  expressly  states  that  the  length  of 
such  lines  as  returned  by  the  surveyor-general  shall  be  held  and  con- 
sidered as  the  true  length  thereof. 


Northwest  Quarter,  Sec.  18. 


FIG.  72. 


10.  The  accompanying  figure 
is  a  copy  of  the  plat  of  the  U.  S. 
survey  of  this  quarter  section. 

A  owns  the  whole  quarter. 
He  sells  to  B  the  W.  1  of  the  N. 
W.  \  of  section  18,  containing 
91^%  acres.  At  about  the  same 
time  he  sells  to  C  the  E.  £  of 
the  N.  W.  i  of  section  18,  con- 
taining 91TVb-  acres. 


MISCELLANEOUS.- 


369 


Where  shall  the  surveyor  run  the  dividing  line  between 
BandC? 

Ans.  The  language  of  the  deed  clearly  shows  the  inten- 
tion of  A  to  sell  and  of  B  and  C  to  purchase  each  the  half 
of  the  area  of  the  quarter  section.  The  surveyor  should 
so  locate  the  line  as  to  carry  out  the  evident  intent  of 
the  parties.  See  rule  2,  p.  244  and  rule  14,  p.  284 .  The  fact 
that  the  quarter  is  differently  subdivided  on  the  govern- 
ment plat  has  no  bearing  on  this  case. 

11.  Certain  early  surveyed  townships  had  three  sets  of 
corners  on  the  range  lines.  (1)  Those  set  when  the  range 
lines  were  run;  (2)  Those  set  as  closing  corners  running 
east;  (3)  Those  set  as  closing  corners  running  west.  What 
use  is  made  of  each  set  of  corners  ? 

Ans.  The  first  corners  set  determine  the  location  of  the 
range  line.  The  second  and  third  sets  of  corners  deter- 
mine the  location  of  their  respective  section  lines  which 
close  on  and  terminate  at  the  range  line. 


FIG.  73. 

12.  This  figure  shows  a  fractional  township  on  the 
Ohio  River.  The  figures  show  the  dimensions  of  section 
1,  as  shown  by  the  field  notes  of  the  United  States  survey. 
By  a  subsequent  measure, 

AB  —  82.25  chains,  and  AD  =  79.50  chains. 


370 


A    MANUAL    OF    LAND    SURVEYING. 


How  shall  the  northeast  quarter  of  section  1  be  laid 
off,  no  quarter-posts  having  been  planted  ? 

Ans.  Place  the  quarter-section  corners  on  the  north  and 
east  sides  of  the  section  in  line  with  and  midway  between 
their  respective  section  corners.  Make  the  east  and  west 
quarter-line  parallel  with  the  south  line  of  the  section, 
placing  the  west  quarter-post  at  the  point  where  the 
quarter-line  thus  run  intersects  the  section  HDC.  From 
the  north  quarter-post  run  the  quarter-line  south  on  a 
course  which  is  a  mean  between  the  courses  of  the  east 
and  the  west  lines  of  the  section,  placing  the  south  quar- 
ter post  at  the  intersection  of  the  section  and  quarter- 
section  lines. 

The  exceptional  features  of  this  case  are  that  no  quar- 
ter-posts were  set  on  the  United  States  survey,  and  that 
the  east  line  of  the  section  is  just  80  chains  in  length, 
having  been  run  from  the  north  to  the  south. 


FIG.  74. 

13.  The  description  in  the  deed  runs:  "  Beginning  at  a 
stone  (A\  at  the  N.W.  corner  of  lot  401;  thence  east  112  ft. 
to  a  stone  (By,  thence  S.  36i°  W.  100  ft.;  thence  west  par- 
allel with  AB  to  the  west  line  of  said  lot  401 ;  thence  north 
on  west  line  of  said  lot,  66  ft.,  to  the  place  of  beginning." 
The  points  A  and  B,  and  angle  ABC,  are  fixed.  C,  by 


MISCELLANEOUS.  3  r  1 

construction  and  in  fact,  is  80^  ft.  distant,  at  right  an- 
gles from  the  line  AS. 

1.  Shall  I  locate  CD  parallel  with  AB,  or  locate  D  66  ft. 
from  A  ? 

2.  Have  I  any  right  to  consider  any  apparent  intention 
to  locate  66  ft.  or  80^  ft.  from  A  ? 

3.  Have  I,  if  I  know  it,  any  authority  to  consider  the 
actual  intention  of  the  grantor  to  locate  CD  V 

4.  If  the  distance  AS  should  actually  measure  114  ft. 
am  I  to  use  it,  or  shall  I  make  J5  112  ft.  from  A  ? 

Ans.  1.  The  answer  to  this  question  will  depend  upon 
the  state  of  facts  brought  out  in  answer  to  questions  2 
and  3.  If  there  be  evidence  showing  what  the  intention 
and  understanding  of  the  parties  to  the  conveyance  was 
as  to  which  of  the  two  lines  should  be  taken,  that  evi- 
dence would  settle  the  question.  If  not,  that  construc- 
tion may  be  given  to  the  deed  which  will  operate  most 
strongly  against  the  grantor  and  give  the  grantee  the 
greater  amount  of  land.  So  far  as  anything  is  shown  in 
the  question,  the  deeds  to  the  adjacent  land  might  fur- 
nish the  necessary  evidence. 

2  and  3.  Yes.  Judge  Cooley  says,  (see  "  Judicial  Func- 
tions of  Surveyors"):  "The  surveyor  must  inquire  into 
all  the  facts,  giving  due  prominence  to  the  acts  of  parlies 
concerned,  and  always  keeping  in  mind  *  *  *  *  that  courts 
and  juries  may  be  required  to  follow  after  the  surveyor 
over  the  same  ground,  and  that  it  is  exceedingly  desirable 
that  he  govern  his  action  by  the  same  lights  and  the  same 
rules  that  will  govern  theirs." 

4.  The  monument  controls  the  distance. 

14.  A  piece  of  land  is  sold,  and  described  as  commencing 
at  the  north  quarter-post  of  section  15,  and  running 
thence  east  100  rods;  thence  south  160  rods;  thence  west 
100  rods;  thence  north  160  rods,  to  the  place  of  beginning; 
containing  100  acres,  according  to  the  United  States  sur- 
vey. 


372  A   MANUAL    OF    LAND    SURVEYING. 

Ques.  How  shall  it  be  set  off  ? 

Ans.  The  deed  clearly  indicates  the  understanding  of 
the  parties  to  the  conveyance  to  be  that  the  land  should 
pass  according  to  the  rules  that  govern  the  United  States 
survey.  One  of  these  rules  is,  that  "the  length  of  the 
boundary  lines  as  returned  by  the  surveyor-general  shall 
be  held  and  considered  as  the  true  length  thereof."  Hence 
in  this  case,  measure  east  from  the  quarter-post  along  the 
section  line  25  chains  of  just  such  measure  as  the  United 
States  surveyors  gave ;  or  in  other  words,  of  pro  rata 
measurement.  Suppose  the  distance  by  the  field  notes  to 
be  40.32  chains  from  quarter-post  to  section  corner.  Then 

25.00 

lay  off of  that  distance.    Proceed  in  a  similar  man- 

40.32 

ner,  running  east  on  the  quarter-line  from  the  center  of 
the  section,  and  the  two  points  thus  located  will  be  the 
corners  of  the  100  acres.  To  get  the  length  of  the  south 
line  of  the  N.  E.  J  of  the  section  by  the  United  States 
survey,  take  the  half  sum  of  the  measure  given  on  the 
north  and  the  south  lines  of  the  section.  Supposing  it  to 
be  40.32  on  the  north,  and  40.18  on  the  south,  then  the 
distance  on  the  quarter-line  would  be  equal  to 

40.32  +  40.18 

=  40.25, 

2 

25.00 

and  you  should  measure  off of  this  distance  for  the 

40.25 
corner. 

15.  A  man  buys  a  tract  of  land  described  as  the 
north  40  acres  of  the  northeast  quarter  of  Section  3. 
This  section  overruns  the  government  measure  when 
measured  with  a  standard  chain. 

Should  this  land  be  measured  as  40  acres  standard 
measure  or  should  the  division  be  made  so  as  to  include 
the  proper  proportion  of  the  overplus? 


MISCELLANEOUS. 


373 


Ans.  In  the  absence  of  evidence  to  prove  a  different 
intention  on  the  part  of  the  parties  to  the  conveyance, 
it  should  be  measured  according  to  the  U.  S.  Govern- 
ment measure  thereof,  as  explained  in  the  answer  to 
the  previous  question. 

16.  "  Section  3  is  fractional  on  Grand  Traverse 
Bay,  the  center  being  a  few  rods  out  in  the  bay.  How 
shall  the  quarter  lines  be  run?"  See  Figure  75. 


•Sec  Line 


FIG.  75. 

Ans.  As  Section  3  is  fractional  in  the  north  half, 
run  the  east  and  west  quarter-line  parallel  with  the 
south  line  of  the  section.  Run  the  north  and  south 
quarter-line  parallel  with  the  west  line  of  the  section. 
The  field  notes  were  not  given  with  the  question.  A 
full  knowledge  of  what  they  contain  might  show 
reason  for  modifying  the  answer. 

17.  A  sells  land  to  B,  described  as  30  acres  off  the 
west  side  of  Lot  1  of  the  section.  Figure  76  repre- 
sents Lot  1  as  shown  on  the  official  plat.  A's  intention 
was  to  sell  30  acres,  and  have  a  strip  left.  Does  the 
surveyor  take  the  government  returns  for  it,  or  does 
he  have  to  run  it  out  following  the  meander  lines,  and 
then  part  off  all  but  the  30  acres,  be  it  more  or  less? 


374  A    MANUAL    OF    LAND    SURVEYING. 

If  the  land  actually  measured  only  30  acres,  would  B 
be  entitled  to  the  whole  of  it  or  only  his  proportion  of 
33.60  acres? 


FIG.    76. 

Ans.  A's  intention  cuts  no  figure  in  the  case,  unless 
it  was  understood  and  shared  in  by  B.  If  there  was  a 
mutual  understanding  and  agreement  between  the 
parties,  it  would  guide  the  surveyor  in  locating  the 
line.  In  the  absence  of  evidence  as  to  the  mutual 
intent  of  the  parties,  B  is  entitled  to  his  proportion 
of  the  lot,  as  shown  by  the  official  plat.  It  has  been 
decided  by  the  supreme  court  of  Michigan  that  in  such 
cases  the  government  meander  line  is  to  be  used  in 
making  the  computation. 

18.  "  How  much  variation  shall  I  allow  for  a 
period  .of,  say  24  years,  in  the  notes  of  the  survey  of 
an  angling  road  of  many  courses,  which  are  as  fol- 
lows," etc.? 

Ans.  I  cannot  tell.  The  annual  change  of  declina- 
tion approximates  4'  in  Michigan.  If  you  can  definite- 
ly locate  any  two  points  of  the  original  survey  of  the 
road,  you  can  run  a  random  line  between  them,  and 
find  the  exact  variation  to  allow.  For  methods  of 
doing  this  where  more  than  one  course  has  to  be  run, 
see  page  79,  and  the  paper  entitled,  "  That  Problem  in 
Land  Surveying,"  in  the  Michigan  Engineers'  Annual 
for  1891,  page  36.  If  there  are  not  two  known  points 
of  the  original  survey  of  the  road,  you  had  better  not 


MISCELLANEOUS. 


375 


try  to  relocate  the  original  line  by  surveys.  It  cannot 
be  done  with  any  certainty,  because  you  have  no 
means  of  comparing  your  compass  and  chain,  or  transit 
and  tape,  with  that  used  on  the  original  survey.  Bet- 
ter make  a  relocation  of  the  road  in  accordance  with 
actual  occupation,  or  such  other  location  as  may  be 
asked  for  by  the  parties  interested. 

19.  Since  the  original  survey  of  Section  9,  the  Mis- 
sissippi River  has  changed  its  course,  adding  by  accre- 
tion nearly  an  entire  section  in  Section  16  and  large 
amounts  in  other  sections.  How  shall  I  survey  the 
accretion  in  Section  16?  The  field  notes  show  that 
the  line  between  Sections  9  and  16  was  run,  and 
meander  posts  and  quarter  post  established. 


FIG.  77. 


876  A    MANUAL    OF    LAND    SURVEYING. 

Ans.  The  official  plat  does  not  show  any  Section  16, 
but  it  does  show  that  the  outlying  fractions,  which 
would  otherwise  have  constituted  a  fractional  Section 
16,  were  attached  to  the  lots  in  Section  9.  Any  accre- 
tions which  have  formed  against  those  lots  belong  to 
the  owners  of  the  lots. 

To  locate  the  boundaries  of  the  lots  on  the  accretion, 
relocate  the  old  river  line  as  far  as  the  accretion  ex- 
tends, and  mark  the  points  where  it  is  intersected  by 
the  lot  lines  as  shown  on  the  official  plat.  Measure 
the  amount  of  the  old  shore  line  which  each  lot  has. 
Then  measure  the  new  shore  line  and  assign  to  each 
lot  the  same  proportion  of  the  new  shore  line  that  it 
has  of  the  old,  and  connect  the  corresponding  points 
on  the  two  shore  lines  by  straight  lines. 

A  strict  carrying  out  of  the  principles  involved 
would  call  for  curved  lines  to  correspond  with  the 
changing  contours  as  the  river  receded,  but  this  is 
hardly  practicable,  and  is  not  required  by  the  courts. 

20.  How  shall  I  run  the  east  and  west  quarter  line 
of  Section  6,  where  the  east  quarter  post  has  not  been 
and  can  not  be  fixed,  but  the  west  quarter  post  and  the 
exterior  lines  of  the  section  are  known? 

Ans.  The  rules  and  regulations  prescribed  by  the 
Secretary  of  the  Interior,  as  provided  by  Section  2397 
of  the  Revised  Statutes  of  the  United  States,  provide 
that  all  the  east  and  west  subdivision  lines  of  these  frac- 
tional sections  adjoining  the  north  boundary  of  the 
township  shall  be  made  parallel  with  the  south  line  of 
the  section,  and  that  in  those  fractional  sections  adjoin- 
ing the  west  boundary  of  the  township,  the  north  and 
south  subdivision  lines  shall  be  made  parallel  with  the 
east  line  of  the  section.  Hence  in  this  case  run  the  quar- 
ter line  east  from  the  west  quarter  post  on  a  line  parallel 
with  the  south  line  of  the  section.  The  rule  for  running 
lines  on  mean  courses  does  not  apply  in  these  cases. 


MISCELLANEOUS.  o  i  i 

21.  What  is  the  penalty  for  destroying  or  removing 
a  "  government  corner  ?" 

Ans.  An  act  of  Congress  approved  June  10,  1896, 
contains  the  following:— 

Provided  further,  That  hereafter  it  shall  be  unlaw- 
ful for  any  person  to  destroy,  deface,  change,  or  remove 
to  another  place,  any  section  corner,  quarter-section 
corner,  or  meander  post,  on  any  government  line  of 
survey,  or  to  cut  down  any  witness  tree  or  any  tree 
blazed  to  mark  the  line  of  the  government  survey,  or 
to  deface  or  remove  any  monument  or  bench  mark  of 
any  government  survey.  That  any  person  who  shall 
offend  against  any  of  the  provisions  of  this  paragraph 
ehall  be  deemed  guilty  of  a  misdemeanor,  and  upon 
conviction  thereof  in  any  'court  shall  be  fined  not  ex- 
ceeding two  hundred  and  fifty  dollars,  or  be  imprisoned 
not  more  than  one  hundred  days.  All  the  fines  accru- 
ing under  this  paragraph  shall  be  paid  into  the  treas- 
ury, and  the  informer,  in  each  case  of  conviction,  shall 
be  paid  the  sum  of  twenty-five  dollars.  (29  Stat.  L., 

343.) 
2.      The  Bights,  Duties  and  Responsibilities  of 

Surveyors. — Surveyors,  by  the  consent  and  acquiescence 
of  the  parties  concerned,  are  usually  the  arbiters  of  dis- 
puted boundaries,  and  their  decisions,  when  thus  acqui- 
esced in  by  the  parties,  become  in  time  as  binding,  and  as 
much  respected  by  the  authorities,  as  the  decisions  of 
juries  and  courts  of  law.  It  is  probable  that  at  least 
ninety-nine  per  cent,  of  all  questions  of  disputed  bound- 
aries are  thus  settled  by  the  interested  parties  themselves, 
in  accordance  with  the  decision  of  the  surveyor. 

Surveyors,  from  constantly  exercising  this  seeming 
authority,  come  at  last  in  many  cases  to  believe  it  to  be 
absolute  and  final,  something  which  must  be  respected, 
overlooking  the  fact  that  the  only  force  their  decisions 
have  comes  from  the  consent  of  the  parties.  When  that 


378  A    MANUAL    OP    LAND    SURVEYING. 

consent  is  withheld,  the  case  goes  to  the  courts  for  settle- 
ment; and  thus  the  courts  have  in  some  cases  felt  called 
upon  to  define  the  surveyor's  standing  before  the  law. 
They  say: 

1.  "  Surveyors  have  no  more  authority  than  other  men 
to  determine  boundaries,  of  their  own  motion.  All  bounds 
and  starting  points  are  questions  of  fact  to  be  determined 
by  testimony.    Surveyors  may  or  may  not  have  in  certain 
cases  means  of  judgment  not  possessed  by  others,  but  the 
law  can  not  and  does. not  make  them  arbiters  of  private 
rights. 

Cronin  v.  Gore,  38  Mich.  381. 

2.  The  law  recognizes  surveyors  as  useful  assistants  in 
doing  the  mechanical  work  of  measurement,  and  calcu- 
lation, and  also  allows  such  credit  to  their  judgment  as 
belongs  to  any  experience  which  may  give  it  value  in 
cases  where  better  means  of  information  do  not  exist. 
But  the  determination  of  facts  belongs  exclusively  to 
courts  and  juries.   Where  a  section  line  or  other  starting 
point  actually  exists,  is  always  a  question  of  fact,  and 
cannot  be  left  to  the  opinion  of  an  expert  for  final  decis- 
ion.   And  where,  as  is  generally  the  case  in  an  old  com- 
munity, boundaries  have  been  fixed  by  long  use  and 
acquiescence,  it  would  be  contrary  to  all  reason  to  have 
them  interfered  with  on  any  abstract  notion  of  science. 

Stewart  v.  Carleton,  31  Mich.  273. 
Gregory  v.  Knight,  50  Mich.  61. 

3.  New  surveys  disturbing  old  boundaries  are  not  to  be 
encouraged. 

Toby  v.  Secor,  Wisconsin.    N.  W.  Reporter,  Vol.  19,  p.  79. 

4.  Lines  long  unquestioned  ought  not  to  be  disturbed 
upon  a  mere  disagreement  among  surveyors,  especially 
when  the  last  survey  is  made  under  the  unfavorable  cir- 
cumstances of  corner  posts  and  witness  trees  being  gone, 
which  it  is  probable  to  suppose  were  in  existence  at  the 
time  of  the  first  survey. 

Case  v.  Trapp,  49  Mich.  59. 


MISCELLANEOUS.  379 

5.  County  surveyors'  certificate    are  not  admissible  in 
evidence  unless  they  contain  all  the  particulars  required 
by  the  statute  to  be  entered  in  the  surveyor's  record. 

Smith  v.  Rich,  37  Mich.  549. 

The  statute  of  Michigan  required  the  length  of  all  lines 
run,  the  area  of  lands  surveyed,  and  other  particulars,  to 
be  entered  in  the  county  surveyor's  record.  In  the  above 
case  the  survey  was  solely  to  find  the  location  of  a  corner 
post.  As  the  surveyor's  certificate  did  not  show  any  area 
of  land  surveyed,  it  was  not  admitted  in  evidence. 

6.  A  surveyor  was  called  on  to  survey  the  line  of  a 
highway.    He  performed  the  work  so  unskillf ully  as  to 
render  a  new  survey  necessary.    A  large  amount  of  road 
constructed  at  great  expense,  on  the  line  designated  by 
the  surveyor  before  the  mistake  was  discovered,  had  to 
be  abandoned.    Action  was  bfrought  to  recover  damages. 
Held,  that  whether  the  defendant  was  a  professional  or 
official  surveyor,  or  represented  himself  as  such,  his  under- 
taking was  that  he  should  bring  to  the  work  the  neces- 
sary knowledge  and  skill  to  perform  the  same  properly 
and  correctly;  and  if  he  failed  to  do  so,  and  the  plaintiff 
suffered  damage  in  consequence  of  such  failure,  the  plain- 
tiff will  be  entitled  to  recover. 

Commissioner  of  Highways  v.  Beebe,  Mich.  Sup.  Court. 
N.  W.  Rep.,  Vol.  20,  No.  16. 

The  following  paper,  by  Chief  Justice  Cooley,  of  the 
Supreme  Court  of  Michigan,  discusses  more  fully  the 
surveyor's  functions : 

3  The  Judicial  Functions  of  Surveyors.— 
When  a  man  has  had  a  training  in  one  of  the  exact 
sciences,  where  every  problem  within  its  purview  is  sup- 
posed to  be  susceptible  of  accurate  solution,  he  is  likely 
to  be  not  a  little  impatient  when  he  is  told  that,  under 
some  circumstances,  he  must  recognize  inaccuracies,  and 
govern  his  action  by  facts  which  lead  him  away  from  the 


380  A    MANUAL    OF    LAND    SURVEYING. 

results  which  theoretically  he  ought  to  reach.  Observa- 
tion warrants  us  in  saying  that  this  remark  may  fre- 
quently be  made  of  surveyors. 

In  the  State  of  Michigan,  all  our  lands  are  supposed  to 
have  been  surveyed  once  or  more,  and  permanent  monu- 
ments fixed  to  determine  the  boundaries  of  those  who 
should  become  proprietors.  The  United  States,  as  orig- 
inal owner,  caused  them  all  to  be  surveyed  once  by  sworn 
officers,  and  as  the  plan  of  subdivision  was  simple,  and 
was  uniform  over  a  large  extent  of  territory,  there  should 
have  been,  with  due  care,  few  or  no  mistakes;  and  long 
rows  of  monuments  should  have  been  perfect  guides  to 
the  place  of  any  one  that  chanced  to  be  missing.  The 
truth  unfortunately  is,  that  the  lines  were  very  carelessly 
run,  the  monuments  inaccurately  placed;  and, as  there- 
corded  witnesses  to  these  were  many  times  wanting  iu 
permanency,  it  is  often  the  case  that  when  the  monument 
was  not  correctly  placed,  it  is  impossible  to  determine  by 
the  record,  by  the  aid  of  anything  on  the  ground,  where 
it  was  located.  The  incorrect  record  of  course  becomes 
worse  than  useless  when  the  witnesses  it  refers  to  have 
disappeared. 

It  is,  perhaps,  generally  supposed  that  our  town  plats 
were  more  accurately  surveyed,  as  indeed  they  should 
have  been,  for  in  general  there  can  have  been  no  difficulty 
in  making  them  sufficiently  perfect  for  all  practical  pur- 
poses. Many  of  them,  however,  were  laid  out  in  the 
woods;  some  of  them  by  .proprietors  themselves,  without 
either  chain  or  compass,  and  some  by  imperfectly  trained 
surveyors,  who,  when  land  was  cheap,  did  not  appreciate 
the  importance  of  having  correct  lines  to  determine 
boundaries  when  land  should  become  dear.  The  fact 
probably  is,  that  town  surveys  are  quite  as  inaccurate  as 
those  made  under  authority  of  the  general  government. 

It  is  now  upwards  of  fifty  years  since  a  major  part  of 
the  public  surveys  in  what  is  now  the  State  of  Michigan 


MISCELLANEOUS.  381 

were  made  under  authority  of  the  United  States.  Of  the 
lands  south  of  Lansing,  it  is  now  forty  years  since  the 
major  part  were  sold,  and  the  work  of  improvement  be- 
gan. A  generation  has  passed  away  since  they  were  con- 
verted into  cultivated  farms,  and  few  if  any  of  the 
original  corner  and  quarter  stakes  now  remain. 

The  corner  and  quarter  stakes  were  often  nothing  but 
green  sticks  driven  into  the  ground.  Stones  might  be 
put  around  or  over  these  if  they  were  handy,  but  often 
they  were  not,  and  the  witness  trees  must  be  relied  upon 
after  the  stake  was  gone.  Too  often  the  first  settlers  were 
careless  in  fixing  their  lines  with  accuracy  while  monu- 
ments remained,  and  an  irregular  brush  fence,  or  some- 
thing equally  untrustworthy,  may  have  been  relied  upon 
to  keep  in  mind  where  the  blazed  line  once  was.  A  fire 
running  through  this  might  sweep  it  away,  and  if  nothing 
was  substituted  in  its  place,  the  adjoining  proprietors 
might  in  a  few  years  be  found  disputing  over  their  lines, 
and  perhaps  rushing  into  litigation,  as  soon  as  they  had 
occasion  to  cultivate  the  land  along  the  boundary. 

If  now  the  disputing  parties  call  in  a  surveyor,  it  is  not 
likely  that  any  one  summoned  would  doubt  or  question 
that  his  duty  was  to  find,  if  possible,  the  place  of  the 
original  stakes  which  determined  the  boundary  line  be- 
tween the  proprietors.  However  erroneous  may  have 
been  the  original  survey,  the  monuments  that  were  set 
must  nevertheless  govern,  even  though  the  effect  be  to 
make  one  half -quarter  section  ninety  acres  and  the  one 
adjoining  seventy ;  for  parties  buy,  or  are  supposed  to 
buy,  in  reference  to  these  monuments,  and  are  entitled  to 
what  is  within  their  lines,  and  no  more,  be  it  more  or  less. 
While  the  witness  trees  remain,  there  can  generally  be  no 
difficulty  in  determining  the  locality  of  the  stakes. 

When  the  witness-trees  are  gone,  so  that  there  is  no 
longer  record  evidence  of  the  monuments,  it  is  remark- 
able how  many  there  are  who  mistake  altogether  the  duty 


382  A    MANUAL    OF    LAND    SURVEYING. 

that  now  devolves  upon  the  surveyor.  It  is  by  no  means 
uncommon  that  we  find  men,  whose  theoretical  education 
is  thought  to  make  them  experts,  who  think  that  when 
the  monuments  are  gone,  the  only  thing  to  be  done  is  to 
place  new  monuments  where  the  old  ones  should  have 
been,  and  would  have  been,  if  placed  correctly.  This  is  a 
serious  mistake.  The  problem  is  now  the  same  that  it 
was  before:  To  ascertain  by  the  best  lights  of  which  the 
case  admits,  where  the  original  lines  were.  The  mistake 
above  alluded  to,  is  supposed  to  have  found  expression  in 
our  legislation;  though  it  is  possible  that  the  real  intent 
of  the  act  to  which  we  shall  refer  is  not  what  is  com- 
monly supposed. 

An  act  passed  in  1869,  (Compiled  Laws,  §  593),  amending 
the  laws  respecting  the  duties  and  powers  of  county  sur- 
veyors, after  providing  for  the  case  of  corners  which  can 
be  identified  by  the  original  field  notes  or  other  unques- 
tionable testimony,  directs  as  follows: 

"  /Second.  Extinct  interior  section  corners  must  be  re-established  at 
the  intersection  of  two  right  lines  joining  the  nearest  known  points  on 
the  original  section  lines  east  and  west  and  north  and  south  of  it. 

•'  Third.  Any  extinct  quarter-section  corner,  except  on  fractional 
lines,  must  be  re-established  equidistant  and  in  a  right  line  between 
the  section  corners ;  in  all  other  cases  at  its  proportionate  distance 
between  the  nearest  original  corners  on  the  same  line." 

The  corners  thus  determined,  the  surveyors  are  required 
to  perpetuate  by  noting  bearing  trees  when  timber  is  near. 

To  estimate  properly  this  legislation,  we  must  start  with 
the  admitted  and  unquestionable  fact  that  each  purchaser 
from  government  bought  such  land  as  was  within  the 
original  boundaries,  and  unquestionably  owned  it  up  to 
the  time  when  the  monuments  became  extinct.  If  the 
monument  was  set  for  an  interior  section  corner,  but  did 
not  happen  to  be  "  at  the  intersection  of  two  right  lines 
joining  the  nearest  known  points  on  the  original  section 
lines  east  and  west  and  north  and  south  of  it,"  it  never- 
theless determined  the  extent  of  his  possessions,  and  he 


MISCELLANEOUS.  383 

gained  or  lost  according  as  the  mistake  did  or  did  not 
favor  him. 

It  will  probably  be  admitted  that  no  man  loses  title  to 
his  land  or  any  part  thereof  merely  because  the  evidences 
become  lost  or  uncertain.  It  may  become  more  difficult 
for  him  to  establish  it  as  against  an  adverse  claimant, 
but  theoretically  the  right  remains;  and  it  remains  as  a 
potential  fact  so  long  as  he  can  present  better  evidence 
than  any  other  person.  And  it  may  often  happen  that 
notwithstanding  the  loss  of  all  trace  of  a  section  corner 
or  quarter  stake,  there  will  still  be  evidence  from  which 
any  surveyor  will  be  able  to  determine  with  almost  abso- 
lute certainty  where  the  original  boundary  was  between 
the  government  subdivisions. 

There  are  two  senses  in  which  the  word  extinct  may  be 
used  in  this  connection:  One,  the  sense  of  physical  dis- 
appearance ;  the  other,  the  sense  of  loss  of  all  reliable 
evidence.  If  the  statute  speaks  of  extinct  corners  in  the 
former  sense,  it  is  plain  that  a  serious  mistake  was  made 
in  supposing  that  surveyors  could  be  clothed  with  author- 
ity to  establish  new  corners  by  an  arbitrary  rule  in  such 
cases.  As  well  might  the  statute  declare  that  if  a  man 
loses  his  deed,,  he  shall  lose  his  land  altogether. 

But  if  by  extinct  corner  is  meant  one  in  respect  to  the 
actual  location  of  which  all  reliable  evidence  is  lost,  then 
the  following  remarks  are  pertinent : 

1.  There  would  undoubtedly  be  a  presumption  in  such 
a  case  that  the  corner  was  correctly  fixed  by  the  govern- 
ment surveyor  where  the  field  notes  indicated  it  to  be. 

2.  But  this  is  only  a  presumption,  and  may  be  over- 
come by  any  satisfactory  evidence  showing-  that  in  fact 
it  was  placed  elsewhere. 

3.  No  statute  can  confer  upon  a  county  surveyor  the 
power  to  "  establish  "  corners,  and  thereby  bind  the  par- 
ties concerned.    Nor  is  this  a  question  merely  of  conflict 
between  State  and'federal  law;  it  is.  a  question  of  prop- 


384  A    MANUAL    OF    LAND    SURVEYING. 

erty  right.  The  original  surveys  must  govern,  and  the 
laws  under  which  they  were  made  must  govern,  because 
the  land  was  bought  in  reference  to  them;  and  any  legis- 
lation, whether  State  or  federal,  that  should  have  the 
effect  to  change  these,  would  be  inoperative,  because  dis- 
turbing vested  rights. 

4.  In  any  case  of  disputed  lines,  unless  the  parties 
concerned  settle  the  controversy  by  agreement,  the  deter- 
mination of  it  is  necessarily  a  judicial  act,  and  it  must 
proceed  upon  evidence,  and  give  full  opportunity  for  a 
hearing.  No  arbitrary  rules  of  survey  or  of  evidence  can 
be  laid  down  whereby  it  can  be  adjudged. 

The  general  duty  of  a  surveyor  in  such  a  case  is  plain 
enough.  He  is  not  to  assume  that  a  monument  is  lost 
until  after  he  has  thoroughly  sifted  the  evidence  and 
found  himself  unable  to  trace  it.  Even  then  he  should 
hesitate  long  before  doing  anything  to  the  disturbance  of 
settled  possessions.  Occupation,  especially  if  long  con- 
tinued, often  affords  very  satisfactory  evidence  of  the 
original  boundary  when  no  other  is  attainable;  and  the 
surveyor  should  inquire  when  it  originated,  how,  and 
why  the  lines  were  then  located  as  they  w.ere,  and  whether 
a  claim  of  title  has  always  accompanied  the  possession, 
and  give  all  the  facts  due  force  as  evidence.  Unfortun- 
ately, it  is  known  that  surveyors  sometimes,  in  supposed 
obedience  to  the  State  statute,  disregard  all  evidences  of 
occupation  and  claim  of  title,  and  plunge  whole  neigh- 
borhoods into  quarrels  and  litigation  by  assuming  to 
"  establish  "  corners  at  points  with  which  the  previous 
occupation  cannot  harmonize.  It  is  often  the  case  that 
where  one  or  more  corners  are  found  to  be  extinct,  all 
parties  concerned  have  acquiesced  in  lines  which  were 
traced  by  the  guidance  of  some  other  corner  or  landmark, 
which  may  or  may  not  have  been  trustworthy;  but  to 
bring  these  lines  into  discredit  when  the  people  concerned 
do  not  question  them,  not  only  breeds  trouble  in  the 
neighborhood,  but  it  must  often  subject  the  surveyor 


MISCELLANEOUS.  335 

himself  to  annoyance  and  perhaps  discredit,  since  in  a 
legal  controversy  the  law  as  well  as  common  sense  must 
declare  that  a  supposed  boundary  line  long  acquiesced  in 
is  better  evidence  of  where  the  real  lino  should  be  than 
any  survey  made  after  the  original  monuments  have  dis- 
appeared. (Stewart  v.  Carleton,  31  Mich.  Reports,  270; 
Diehl  v.  Zanger,  39  Mich.  Reports,  601.)  And  county  sur- 
veyors, no  more  than  any  others,  can  conclude  parties  by 
their  surveys. 

The  mischiefs  of  overlooking  the  facts  of  possession 
most  often  appear  in  cities  and  villages.  In  towns  the 
block  and  lot  stakes  soon  disappear;  there  are  no  witness 
trees,  and  no  monuments  to  govern  except  such  as  have 
been  put  in  their  places,  or  where  their  places  were  sup- 
posed to  be.  The  streets  are  likely  to  be  soon  marked  oft 
by  fences,  and  the  lots  in  a  block  will  be  measured  off 
from  these,  without  looking  farther.  Now  it  may  per- 
haps be  known  in  a  particular  case  that  a  certain  monu- 
ment still  remaining  was  the  starting  point  in  the  original 
survey  of  the  town  plat ;  or  a  surveyor  settling  in  the 
town  may  take  some  central  point  as  the  point  of  depart- 
ure in  his  surveys,  and  assuming  the  original  plat  to  be 
accurate,  he  will  then  undertake  to  find  all  streets  and 
all  lots  by  course  and  distance  according  to  the  plat, 
measuring  and  estimating  from  his  point  of  departure. 
This  procedure  might  unsettle  every  line  and  every  mon- 
ument existing  by  acquiescence  in  the  town;  it  would  be 
very  likely  to  change  the  lines  of  streets,  and  raise  con- 
troversies everywhere.  Yet  this  is  what  is  sometimes 
done;  the  surveyor  himself  being  the  first  person  to  raise 
the  disturbing  questions. 

Suppose,  for  example,  a  particular  village  street  has 
been  located  by  acquiescence  and  used  for  many  years, 
and  the  proprietors  in  a  certain  block  have  laid  off  their 
lots  in  reference  to  this  practical  location.  Two  lot  own- 
ers quarrel,  and  one  of  them  calls  in  a  surveyor,  that  he 


386  A    MANUAL    OF    LAND    SURVEYING. 

may  make  sure  his  neighbor  shall  not  get  an  inch  of  land 
from  him.  This  surveyor  undertakes  to  make  his  survey 
accurate,  whether  the  original  was  so  or  not,  and  the  first 
result  is,  he  notifies  the  lot  owners  that  there  is  error  in 
the  street  line,  and  that  all  fences  should  be  moved,  say 
one  foot  to  the  east.  Perhaps  he  goes  on  to  drive  stakes 
through  the  block  according  to  this  conclusion.  Of 
course,  if  he  is  right  in  doing  this,  all  lines  in  the  village 
will  be  unsettled;  but  we  will  limit  our  attention  to  the 
single  block.  It  is  not  likely  that  the  lot  owners  gener- 
ally will  allow  the  new  survey  to  unsettle  their  posses- 
sions, but  there  is  always  a  probability  of  finding  some 
one  disposed  to  do  so.  We  shall  then  have  a  lawsuit; 
and  with  what  result  ? 

It  is  a  common  error  that  lines  do  not  become  fixed  by 
acquiescence  in  a  less  time  than  twenty  years.  In  fact, 
by  statute,  road  lines  may  become  conclusively  fixed  in 
ten  years;  and  there  is  no  particular  time  that  shall  be 
required  to  conclude  private  owners,  where  it  appears 
that  they  have  accepted  a  particular  line  as  their  bound- 
ary, and  all  concerned  have  cultivated  and  claimed  up  to 
it  Public  policy  requires  that  such-  lines  be  not  lightly 
disturbed,  or  disturbed  at  all  after  the  lapse  of  any  con- 
siderable time.  The  litigant,  therefore,  who  in  such  a 
case  pins  his  faith  on  the  surveyor,  is  likely  to  suffer  for 
his  reliance,  and  the  surveyor  himself  to  be  mortified  by 
a  result  that  seems  to  impeach  his  judgment. 

Of  course  nothing  in  what  has  been  said  can  require  a 
surveyor  to  conceal  his  own  judgment,  or  to  report  the 
facts  one  way  when  he  believes  them  to  be  another.  He 
has  no  right  to  mislead,  and  he  may  rightfully  express 
his  opinion  that  an  original  monument  was  at  one  place, 
when  at  the  same  time  he  is  satisfied  that  acquies- 
cence has  fixed  the  rights  of  parties  as  if  it  were  at  an- 
other. But  he  would  do  mischief  if  he  were  to  attempt 
to  "establish"  monuments  which  he  knew  would  tend 
to  disturb  settled  rights;  the  farthest  he  has  a  right  to 


MISCELLANEOUS.  387 

go,  as  an  officer  of  the  law,  is  to  express  his  opinion  where 
the  monument  should  be,  at  the  same  time  that  he  im- 
parts the  information  to  those  who  employ  him,  and  who 
mignt  otherwise  be  misled,  that  the  same  authority  that 
makes  him  an  officer  and  entrusts  him  to  make  surveys, 
also  allows  parties  to  settle  their  own  boundary  lines,  and 
considers  acquiescence  in  a  particular  line  or  monument, 
for  any  considerable  period,  as  strong  if  not  conclusive 
evidence  of  such  settlement.  The  peace  of  the  commu- 
nity absolutely  requires  this  rule.  It  is  not  long  since, 
that  in  one  of  the  leading  cities  of  the  State  an  attempt 
was  made  to  move  houses  two  or  three  rods  into  a  street, 
on  the  ground  that  a  survey  under  wnich  the  street  had 
been  located  for  many  years,  had  been  found  on  a  more 
recent  survey  to  be  erroneous. 

From  the  foregoing,  it  will  appear  that  the  duty  of  the 
surveyor  where  boundaries  are  in  dispute  must  be  varied 
by  the  circumstances.  1.  He  is  to  search  for  original 
monuments,  or  for  the  places  where  they  were  originally 
located,  and  allow  these  to  control  if  he  finds  them,  unless 
he  has  reason  to  believe  that  agreements  of  the  parties, 
express  or  implied,  have  rendered  them  unimportant.  By 
monuments  in  the  case  of  government  surveys  we  mean 
of  course  the  corner  and  quarter-stakes;  blazed  lines  or 
marked  trees  on  the  lines  are  not  monuments:  they  are 
merely  guides  or  finger  posts,  if  we  may  use  the  expres- 
sion, to  inform  us  with  more  or  less  accuracy  where  the 
monuments  may  be  found.  2.  If  the  original  monuments 
are  no  longer  discoverable,  the  question  of  location  be- 
comes one  of  evidence  merely.  It  is  merely  idle  for  any 
State  statute  to  direct  a  surveyor  to  locate  or  "  establish  " 
a  corner,  as  the  place  of  the  original  monument,  accord- 
ing to  some  inflexible  rule.  The  surveyor,  on  the  other 
hand,  must  inquire  into  all  the  facts:  giving  due  promi- 
nence to  the  acts  of  parties  concerned,  and  always  keep- 
ing in  mind,  first,  that  neither  his  opinion  nor  his  survey 


388  A   MANUAL    OF    LAND    SURVEYING. 

can  be  conclusive  upon  parties  concerned;  and,  second, 
that  courts  and  juries  may  be  required  to  follow  after  the 
surveyor  over  the  same  ground,  and  that  it  is  exceedingly 
desirable  that  he  govern  his  action  by  the  same  lights 
and  the  same  rules  that  will  govern  theirs. 

It  is  always  possible,  when  corners  are  extinct,  that  the 
surveyor  may  usefully  act  as  a  mediator  between  parties, 
and  assist  in  preventing  legal  controversies  by  settling 
doubtful  lines.  Unless  he  is  made  for  this  purpose  an 
arbitrator  by  legal  submission,  the  parties,  of  course,  even 
if  they  consent  to  follow  his  judgment,  cannot  on  the 
basis  of  mere  consent,  be  compelled  to  do  so;  but  if  he 
brings  about  an  agreement,  and  they  carry  it  into  effect 
by  actually  conforming  their  occupation  to  his  lines,  the 
action  will  conclude  them.  Of  course,  it  is  desirable  that 
all  such  agreements  be  reduced  to  writing;  but  this  is  not 
absolutely  indispensable  if  they  are  carried  into  effect 
without. 

Meander  Lines.— The  subject  to  which  allusion  will 
now  be  made,  is  taken  up  with  some  reluctance,  because 
it  is  believed  the  general  rules  are  familiar.  Nevertheless, 
it  is  often  found  that  surveyors  misapprehend  them,  or 
err  ia  their  application;  and  as  other  interesting  topics 
are  somewhat  connected  with  this,  a  little  time  devoted 
to  it  will  probably  not  be  altogether  lost.  The  subject  is 
that  of  meander  lines.  These  are  lines  traced  along  the 
shores  of  lakes,  ponds,  and  considerable  rivers,  as  the 
measures  of  quantity  when  sections  are  made  fractional 
by  such  waters.  These  have  determined  the  price  to  be 
paid  when  government  lands  were  bought,  and  perhaps 
the  impression  still  lingers  in  some  minds  that  the  mean- 
der lines  are  boundary  lines,  and  that  all  in  front  of  them 
remains  unsold.  Of  course  this  is  erroneous.  There  was 
never  any  doubt  that,  except  on  the  large  navigable 
rivers,  the  boundary  of  the  owners  of  the  banks  is  the 
middle  line  of  the  river;  and  while  some  courts  have  held 
that  this  was  the  rule  on  all  fresh-water  streams,  large 


MISCELLANEOUS.  389 

and  small,  others  have  held  to  the  doctrine  that  the  title 
to  the  "bed  of  the  stream  below  low-water  mark  is  in  the 
State,  while  conceding  to  the  owners  of  the  banks  all 
riparian  rights.  The  practical  difference  is  not  very  im- 
portant. In  this  State,  the  rule  that  the  center  line  is  the 
boundary  line,  is  applied  to  all  our  great  rivers,  including 
the  Detroit,  varied  somewhat  by  the  circumstance  of 
there  being  a  distinct  channel  for  navigation,  in  some 
cases,  with  the  stream  in  the  main  shallow,  and  also 
sometimes  by  the  existence  of  islands. 

The  troublesome  questions  for  surveyors  present  them- 
selves when  the  boundary  line  between  two  contiguous 
estates  is  to  be  continued  from  the  meander  line  to  the 
center  line  of  the  river.  Of  course,  the  original  survey 
supposes  that  each  purchaser  of  land  on  the  stream  has 
a  water  front  of  the  length  shown  by  the  field  notes;  and 
it  is  presumable  that  he  bought  this  particular  land  be- 
cause of  that  fact.  In  many  cases  it  now  happens  that 
the  meander  line  is  left  some  distance  from  the  shore  by 
the  gradual  change  of  course  of  the  stream,  or  diminu- 
tion of  the  flow  of  water.  Now  the  dividing  line  be- 
tween two  government  subdivisions  might  strike  the 
meander  line  at  right  angles,  or  obliquely;  and,  in  some 
cases,  if  it  were  continued  in  the  same  direction  to  the 
center  line  of  the  river,  might  cut  off  from  the  water  one 
of  the  subdivisions  entirely,  or  at  least  cut  it  off  from 
any  privilege  of  navigation,  or  other  valuable  use  of  the 
water,  while  the  other  might  have  a  water  front  much 
greater  than  the  length  of  a  line  crossing  it  at  right 
angles  to  its  side  lines.  The  effect  might  be  that,  of  two 
government  subdivisions  of  equal  size  and  cost,  one 
would  be  of  very  great  value  as  water-front  property, 
and  the  other  comparatively  valueless.  A  rule  which 
would  produce  this  result  would  not  be  just,  and  it  has 
not  been  recognized  in  the  law. 

Nevertheless  it  is  not  easy  to  determine  what  ought  to 


390  A    MANUAL    OF    LAND    SURVEYING. 

be  the  correct  rule  for  every  case.  If  the  river  has  a 
straight  course,  or  one  nearly  so,  every  man's  equities 
will  be  preserved  by  this  rule:  Extend  the  line  of  divi- 
sion between  the  two  parcels  from  the  meander  line  to 
the  center  line  of  the  river,  as  nearly  as  possible  at  right 
angles  to  the  general  course  of  the  river  at  that  point. 
This  will  preserve  to  each  man  the  water  front  which  the 
field  nctes  indicated,  except  as  changes  in  the  water  may 
have  affected  it,  and  the  only  inconvenience  will  be  that 
the  division  line  between  different  subdivisions  is  likely 
to  be  more  or  less  deflected  where  it  strikes  the  meander 
line. 

This  is  the  legal  rule,  and  is  not  limited  to  government 
surveys,  but  applies  as  well  to  water  lots  which  appear  as 
such  on  town  plats.  '(Bay  City  Gas  Light  Co.  v.  The  In- 
dustrial Works,  28  Mich.  Reports,  182.)  It  often  happens, 
therefore,  that  the  lines  of  city  lots  bounded  on  naviga- 
ble streams  are  deflected  as  they  strike  the  bank,  or  the 
line  where  the  bank  was  when  the  town  was  first  laid  out. 

When  the  stream  is  very  crooked,  and  especially  if  there 
are  short  bends,  so  that  the  foregoing  rule  is  incapable  of 
strict  application,  it  is  sometimes  very  difficult  to  deter- 
mine what  shall  be  done;  and  in  many  cases  the  surveyor 
may  be  under  the  necessity  of  working  out  a  rule  for 
himself.  Of  course  his  action  cannot  be  conclusive;  but 
if  he  adopts  one  that  follows  as  nearly  as  the  circum- 
stances will  admit,  the  general  rule  above  indicated,  so  as 
to  divide  as  near  as  may  be  the  bed  of  the  stream  among 
the  adjoining  owners  in  proportion  to  their  lines  upon 
the  shore,  his  division,  being  that  of  an  expert,  made  upon 
the  ground  and  with  all  available  lights,  is  likely  to  be 
adopted  as  law  for  the  case.  Judicial  decisions,  into 
which  the  surveyor  would  find  it  prudent  to  look  under 
such  circumstances,  will  throw  light  upon  his  duties  and 
may  constitute  a  sufficient  guide  when  peculiar  cases 
arise.  Each  riparian  lot  owner  ought  to  have  a  line  on 


MISCELLANEOUS.  391 

the  legal  boundary,  namely,  the  center  line  of  the  stream 
proportioned  to  the  length  of  his  line  on  the  shore  and 
the  problem  in  each  case  is,  how  this  is  to  be  given  him. 
Alluvion,  when  a  river  imperceptibly  changes  its  course, 
will  be  apportioned  by  the  same  rules. 

The  existence  of  islands  in  a  stream  when  the  middle 
line  constitutes  a  boundary,  will  not  affect  the  apportion- 
ment unless  the  islands  were  surveyed  out  as  government 
subdivisions  in  the  original  admeasurement.  Wherever 
that  was  the  case,  the  purchaser  of  the  island  divides  the 
bed  of  the  stream  on  each  side  with  the  owner  of  the 
bank,  and  his  rights  also  extend  above  and  below  the 
solid  ground,  and  are  limited  by  the  peculiarities  of  the 
bed  and  the  channel.  If  an  island  was  "not  surveyed  as  a 
government  subdivision  previous  to  the  sale  of  the  bank, 
it  is  of  course  impossible  to  do  this  for  the  purposes  of 
government  sale  afterward,  for  the  reason  that  the  rights 
of  the  bank  owners  are  fixed  by  their  purchase;  when 
making  that  they  have  a  right  to  understand  that  all  land 
between  the  meander  lines,  not  separately  surveyed  and 
sold,  will  pass  with  the  shore  in  the  government  sale: 
and  having  this  right,  anything  which  their  purchase 
would  include  under  it  cannot  afterward  be  taken  from 
them.  It  is  believed,  however  that  the  federal  courts 
would  not  recognize  the  applicability  of  this  rule  to  large 
navigable  rivers,  such  as  those  uniting  the  great  lakes. 

On  all  the  little  lakes  of  the  state  which  are  mere  ex- 
pansions near  their  mouths  of  the  rivers  passing  through 
them — such  as  the  Muskegon  Pere  Marquette  and  Manis- 
tee — the  same  rule  of  bed  ownership  has  been  judicially 
applied  that  is  applied  to  the  rivers  themselves;  and  the 
division  lines  are  extended  under  the  water  in  the  same 
way.  (Rice  v.  Euddiman,  10  Mich.,  125.)  If  such  a  lake 
were  circular,  the  lines  would  converge  to  the  center;  if 
oblong  or  irregular,  there  might  be  a  line  in  the  middle 
on  which  they  would  terminate,  whose  course  would  bear 


392  A    MANUAL    OF    LAND    SURVEYING. 

some  relation  to  that  of  the  shore.  But  it  can  seldom  be 
important  to  follow  the  division  line  very  far  under  the 
water,  since  all  private  rights  are  subject  to  the  public 
rights  of  navigation  and  other  use,  and  any  private  use 
of  the  lands  inconsistent  with  these  would  be  a  nuisance, 
and  punishable  as  such.  It  is  sometimes  important,  how- 
ever, to  run  the  lines  out  for  considerable  distance,  in 
order  to  determine  where  one  may  lawfully  moor  vessels 
or  rafts,  for  the  winter,  or  cut  ice.  The  ice  crop  that 
forms  over  a  man's  land  of  course  belongs  to  him.  (Lor- 
man  v.  Benson,  8  Mich.,  18;  People's  Ice  Co.  o.  Steamer 
Excelsior,  recently  decided.) 

What  is  said  above  will  show  how  unfounded  is  the 
notion,  which  is  sometimes  advanced,  that  a  riparian 
proprietor  on  a  meandered  river  may  lawfully  raise  the 
water  in  the  stream  without  liability  to  the  proprietors 
above,  provided  he  does  not  raise  it  so  that  it  overflows 
the  meander  line.  The  real  fact  is  that  the  meander  line 
has  nothing  to  do  with  such  a  case,  and  an  action  will  lie 
whenever  he  sets  back  the  water  upon  the  proprietor 
above,  whether  the  overflow  be  below  the  meander  lines 
or  above  them. 

As  regards  the  lakes  and  ponds  of  the  state,  one  may 
easily  raise  questions  that  it  would  be  impossible  for  him 
to  settle]  Let  us  suggest  a  few  questions,  some  of  which 
are  easily  answered,  and  some  not: 

1.  To  whom  belongs  the  land  under  these  bodies  of 
water,  where  they  are  not  mere  expansions  of  a  stream 
flowing  through  them  ? 

2.  What  public  rights  exist  in  them  ? 

3.  If  there  are  islands  in  them  which  were  not  sur- 
veyed out  and  sold  by  the  United  States,  can  this  be  done 
now? 

Others  wii.  ^e  suggested  by  the  answers  give~  to  these. 

It  seems  obvious  that  the  rules  of  private  ownership 

which  are  applied  to  rivers  cannot  be  applied  to  the  great 


MISCELLANEOUS.  393 

lakes.  Perhaps  it  should  be  held  that  the  boundary  is  at 
low  water  mark,  but  improvements  beyond  this  would 
only  becoifte  unlawful  when  they  became  nuisances 
Islands  in  the  great  lakes  would  belong  to  the  United 
States  until  sold,  and  might  be  surveyed  and  measured 
for  sale  at  any  time.  The  right  to  take  fish  in  the  lakes, 
or  to  cut  ice,  is  public  like  the  right  of  navigation,  but  is 
to  be  exercised  in  such  manner  as  not  to  interfere  with 
the  rights  of  shore  owners.  But  so  far  as  these  public 
rights  can  be  the  subject  of  ownership,  they  belong  to 
the  state,  not  to  the  United  States;  aod  so,  it  is  believed, 
does  the  bed  of  a  lake  also.  (Pollord  v.  Hagan,  3  Howard's 
U.  S.  Keports.)  But  such  rights  are  not  generally  consid- 
ered proper  subjects  of  sale,  but  like  the  right  to  make 
use  of  the  public  highways,  they  are  held  by  the  state  ID 
trust  for  all  the  people. 

What  is  said  of  the  large  lakes  may  perhaps  be  said  also 
of  many  of  the  interior  lakes  of  the  state;  such,  for  ex- 
ample, as  Houghton,  Higgins,  Cheboygan,  Burt's,  MoBet. 
Whitmore,  and  many  others.  But  there  are  many  little 
lakes  or  ponds  which  are  gradually  disappearing,  and  the 
shore  proprietorship  advances  part  passu  as  the  waters 
recede.  If  these  are  of  any  considerable  size — say,  even 
a  mile  across— there  may  be  questions  of  conflicting 
rights  which  no  adjudication  hitherto  made  could  settle 
Let  any  surveyor,  lor  example,  take  the  case  of  a  pond  ot 
irregular  form,  occupying  a  mile  square  or  more  of  terri- 
tory, and  undertake  to  determine  the  rights  of  the  shore 
proprietors  to  its  bed  when  it  shall  totally  disappear,  anrt 
he  will  find  he  is  in  the  midst  of  problems  such  as  proba- 
bly he  has  never  grappled  with,  or  reflected  upon  before^ 
But  the  general  rules  for  the  extension  of  shore  lines, 
which  have  already  been  laid  down,  should  govern  such 
cases,  or  at  least  should  serve  as  guides  in  their  settle- 
ment. 

Where  a  pond  is  so  small  as  to  be  included  within  the 
lines  of  a  private  purchase  from  the  government,  it  is  not 


394  A    MANUAL    OF    LAND    SURVEYING. 

believed  the  public  have  have  any  rights  in  it  whatever. 
Where  it  is  not  so  included,  it  is  believed  they  have  rights 
cf  fishery,  rights  to  take  ice  and  water,  and  rights  of  nav- 
igation for  business  or  pleasure.  This  is  the  common 
belief,  and  probably  the  just  one.  Shore  rights  must  not 
be  so  exercised  as  to  disturb  these,  and  the  states  may 
pass  all  proper  laws  for  their  protection.  It  would  be 
easy  with  suitable  legislation  to  preserve  these  little 
bodies  of  water  as  permanent  places  of  resort  for  the 
pleasure  and  recreation  of  the  people,  and  there  ought  to 
be  such  legislation. 

If  the  state  should  be  recognized  as  owner  of  the  beds 
of  these  small  lakes  and  ponds,  it  would  not  be  owner  for 
the  purpose  of  selling.  It  would  be  owner  only  as  trustee 
for  the  public  use;  and  a  sale  would  be  inconsistent  with 
the  right  of  the  bank  owners  to  make  use  of  the  water  in 
its  natural  condition  in  connection  with  their  estates. 
Some  of  them  might  be  made  salable  lands  by  draining; 
but  the  state  could  not  drain,  even  for  this  purpose, 
against  the  will  of  the  shore  owners,  unless  their  rights 
were  appropriated  and  paid  for. 

Upon  many  questions  that  might  arise  between  the 
state  as  owner  of  the  bed  of  a  little  lake  and  the  shore 
owners,  it  would  be  presumptuous  to  express  an  opinion 
now,  and  fortunately  the  occasion  does  not  require  it. 

I  have  thus  indicated  a  few  of  the  questions  with  which 
surveyors  may  now  and  then  have  occasion  to  deal,  and 
to  which  they  should  bring  good  sense  and  sound  judg- 
ment. Surveyors  are  not  and  cannot  be  judicial  officers, 
but  in  a  great  many  cases  they  act  in  a  quasi  judicial 
capacity  with  the  acquiescence  of  parties  concerned;  and 
it  is  important  for  them  to  know  by  what  rules  they  are 
to  be  guided  in  the  discharge  of  their  judicial  functions. 
What  I  have  said  cannot  contribute  much  to  their  en- 
lightenment, but  I  trust  will  not  be  wholly  without 
value. 


LEVELING.  395 

CHAPTER  XIII. 

BEVELING    AND    DRAINAGE    SURVEYING. 

1.  Leveling  is  the  operation  of  measuring  the  differ- 
ence in  height  of  two  or  more  points. 

The  surface  of  water  at  rest  is  an  example  of  a  level 
surface. 

If  the  earth  was  a  perfect  sphere,  a  line  of  true 
level  would  be  an  arc  or  a  circle  having  its  centre  at  the 
centre  of  gravity  of  the  earth.  So  far  as  common  level- 
ing is  concerned  it  may  be  so  considered,  as  the  error 
arising  therefrom  is  so  small  as  to  be  of  no  practical 
consequence. 

The  line  of  apparent  level  is  a  straight  horizontal 
line  passing  through  the  point  of  observation,  tangent 
to  the  line  of  true  level. 

In  precise  leveling  the  difference  between  true  and 
apparent  level  is  measured,  the  instruments  used  are  of 
the  best,  and  all  the  operations  are  performed  so  as  to 
reduce  the  error  to  the  smallest  possible  amount.  In 
common  leveling  for  streets,  railroads,  drains,  water 
powers  and  the  like  operations,  a  lower  degree  of  ac- 
curacy is  required  and  the  refinements  of  precise  level- 
ing are  dispensed  with."  No  attention  is  paid  to  the  dif- 
ference between  true  and  apparent  level,  it  being  too 
small  to  affect  the  practical  result. 

2.  The  deviation  of  the  true  from  the  apparent  level 
between  two  points  is  equal  to  the  square  of  the  distance 
between  the  points t  divided  by  the  diameter  of  the  earth. 

Also,  The  deviations  for  different  distances  are  pro- 
portional to  the  squares  of  the  distances. 

Calling  the  diameter  of  the  earth  7920  miles  and  ta- 
king points  one  mile  apart,  we  find  the  deviation  = 
0.000126  miles  ==  0.665  ft.  =  7.98  inches.  For  m  miles, 
deviation  =  7.98  m2  inches. 


396  A    MANUAL    OF    LAND    SURVEYING. 

The  effect  of  the  refraction  of  light  is  to  apparently 
increase  the  difference  between  true  and  apparent  level. 

For  considerable  distances  the  correction  for  curvature 
as  above  found  is  sometimes  diminished  by  about  one- 
sixth  of  itself. 

If  the  instrument  is  placed  midway  between  the  points 
whose  difference  in  height  is  required,  the  errors  are  bal- 
anced and  eliminate  each  other,  giving  a  correct  result. 

3.  In  leveling,  two  instruments  are  required,  one  to 
find  a  horizontal  line,  and  the  other  to  measure  vertical 
distances.  These  instruments  are  called  a  Level  and  a 
Leveling  rod. 

Level  lines,  for  many  common  purposes,  on  a  limited 
area,  when  no  instruments  are  at  hand,  can  be  obtained 
by  the  following  method :  — 

Suspend  from  some  fixed  point  of  support  P  by  stout 
cords  as  indicated,  a  pole  of  any  shape  A  B,  having  the 
longer  end  sharpened  to  a  fine  point. 
From  this  pole  hang  a  heavy  weight  R 
s-     as  shown.     Set  two  stakes  S&  so  that 
|A        db      JlL     the    point    of    the    pole   when    swung 
around  will  just  touch  them.    Smooth 
a  place  on  each  stake  to  receive  marks. 
After  taking  the  twist  out  of  the  supporting  cord,  care- 
fully swing  the  pole  around  and  mark  the  exact  place 
where  the  point  of  the  pole  touches  each  stake.    Repeat 
this,  and  take  the  most  satisfactory  points.     They  will 
determine  a  level  line  of  sight. 

A  cheap  instrument  which  almost  any  one  can  make, 
having  a  more  extended  range,  is  made  as  follows :  Take 
two  pieces  of  glass  tubing  three  or  four 
"^     inches  long  and  connect  them  with  a 
rubber  tube  two  or  three  feet  long,  so 
as  to  make  a  continuous  water  tight 
tube,  with  glass  ends.     Pass  the  ends 
of  the  tube  through  holes  in  a  cross  bar 
FIG.  76.          made  of  a  piece  of  board  of  suitable 


LEVELING.  397 

size,  as  shown  in  the  cut,  and  fasten  them  with  the  tops 
projecting  an  inch  or  more  above  the  bar.  The  cross  bar 
may  be  fastened  with  a  bolt  and  nut  to  a  staff  so  that  it 
may  be  set  up  and  adjusted  to  a  level  line.  Colored  fluid 
is  poured  into  the  tube.  The  surface  of  the  fluid  in  the 
glass  tubes  determines  the  level  line.  Sights  of  horse 
hair  or  fine  wire  may  be  attached  close  to  the  glass  tubes 
and  the  cross  bar  adjusted  to  bring  them  into  a  level  line. 

An  instrument  can  thus  be  made  at  the  expense  of  a 
few  cents  in  money  and  a  few  minutes'  labor  that  will  do 
very  satisfactory  work.  ..-?•' 

4.  If  a  tube  be  nearly  filled  with  any  liquid,  as  water, 
alcohol  or  ether,  and  closed,  the  liquid  will  seek  the  lowest 
part,  and  the  vacant  space  or  bubble,  as  it  is  called,  will  be 
found  at  the  highest  part  of  the  tube.  If  the  tube  is  of 
glass,  and  very  truly  ground  on  the  inside  to  a  segment  of 
a  circle,  it  furnishes  the  best  known  means  for  determin- 
ing a  level  line.  Such  tubes  are  made  and  nearly  filled 
with  ether  or  alcohol,  leaving  a  small  space  or  bubble. 
When  such  a  tube  is  placed  convex  side  uppermost,  the 
bubble  seeks  the  highest  point.  Then  a  vertical  line 
passing  through  the  center  of  the  bubble  will  coincide 
with  the  radius  of  the  arc  to  which  the  tube  is  ground.  A 
perpendicular  to  this  vertical  line  is  a  line  of  apparent 
level.  Such  a  tube  is  the  most  essential  part  of  the  level. 
It  is  encased  in  a  brass  tube,  having  an  opening  so  that 
the  bubble  and  as  much  of  the  glass  tube  as  necessary 
can  be  seen.  A  graduated  scale  is  attached  to  it,  or 
marked  on  the  tube,  by  means  of  which  the  bubble  is 
measured  and  its  position  with  relation  to  other  parts  of 
the  instrument  is  determined.  The  tube  thus  prepared  is 
attached  to  a  telescope  in  such  a  manner  that  it  can  be 
adjusted  so  as  to  bring  the  radius  of  the  ground  glass 
perpendicular  to  the  line  of  sight  in  the  telescope.' 

The  telescope  is  mounted  in  such  a  manner  as  to  permit 
it  to  revolve  freely  in  a  horizontal  plane  and  to  be  readily 
adjusted  to  the  line  of  apparent  level. 


A    MANUAL    OF    LAND    SURVEYING. 


FIG.  77. 

The  plan  of  mounting  the  telescope  most  in  favor  in 
the  United  States  is  by  a  horizontal  bar  with  forked 
arms  called  wyes.  The  telescope  rests  upon  the  wyes 
and  is  held  in  place  by  clips  which  may  be  loosened,  per- 
mitting the  telescope  to  be  rolled  over  in  the  wyes.  The 
bar  is  connected  by  a  spindle  to  the  tripod  socket  and 
leveling  head  similar  to  that  used  upon  the  transit.  By 
permission  of  Messrs.  Buff  &  Berger,  of  Boston,  the 
following  quotation  is  taken  from  their  catalogue: 

5.  "  The  Adjustments.  —  In  a  theoretically  perfect 
level  the  following  points  are  established: 

1.  The  object  and  eye-glasses  are  perpendicular  to  the 
optical  axis  at  all  distances  apart. 

2.  The  optical  axis  coincides  with  the  axis  of  rotation 
in  the  wyes. 

3.  The  axis  of  collimation  coincides  with  the  optical 
axis. 

4.  The  axis  of  collimation  is  parallel  to  the  telescope 
level. 

5.  The  collars  resting  in  the  wyes  are  circles  of  the 
same  diameter  and  concentric  with  the  line  of  collima- 
tion of  the  telescope. 

6.  The  wyes  are  exactly  similar,  and  similarly  placed 
with  reference  to  the  line  of  collimation  of  the  telescope. 

7.  The  level  bubble  moves  over  equal  spaces  for  equal 
displacements  of  the  telescope  in  altitude. 


LEVELING.  399 

8.  The  level  bubble  expands  or  contracts  equally  from 
the  centre  in  both  directions,  during  changes  of  tempera- 
ture. 

9.  The  vertical  axis  of  revolution  is  perpendicular  to 
the  line  of  collimation  of  the  telescope. 

Of  the  above,  the  maker  establishes  points  numbered  1, 
2,  5,  6,  7  and  8.  The  remaining  points,  3,  4  and  9,  are 
established  when  the  instrument  leaves  the-  shop,  but 
being  liable  to  derangement  from  rough  usage,  they  are 
made  adjustable  in  the  field. 

Adjusting.  After  the  engineer  has  set  up  the  instru- 
ment and  adjusted  the  eye-piece  for  parallax,  the  hor- 
izontal cross-line  had  better  be  made  to  lie  in  the  plane 
of  the  azimuthal  rotation  of  the  instrument.  This  may 
be  accomplished  by  rotating  the  reticule,  after  loosening 
the  capstan-headed  screws,  until  a  point  remains  bisected 
throughout  the  length  of  the  line  when  the  telescope  is 
moved  in  azimuth.  In  making  this  adjustment,  the  level 
tube  is  to  be  kept  directly  beneath  the  telescope  tube. 
When  made,  the  small  set  screw  attached  to  one  of  the 
wyes  may  be  set  so  that  by  simply  bringing  the  project- 
ing pin  from  the  telescope  against  it,  the  cross-lines  will 
be  respectively  parallel  and  perpendicular  to  the  motion 
of  the  telescope  in  azimuth. 

The  first  collimating  of  the  instrument  may  be  made, 
using  an  edge  of  some  building,  or  any  profile  which  is 
vertical.  Make  the  vertical  cross-line  tangent  to  any 
such  profile,  and  then  turn  the  telescope  half-way  round 
in  its  wyes.  If  the  vertical  cross-line  is  still  tangent  to 
the  edge  selected,  the  vertical  cross-line  is  collimated. 

Select  some  horizontal  line,  and  cause  the  horizontal 
cross-line  to  be  brought  tangent  to  it.  Again  rotate  the 
telescope  half  way  round  in  its  wyes,  and  if  the  hori- 
zontal cross-line  is  still  tangent  to  the  edge  selected,  the 
horizontal  cross-line  is  collimated. 

Having  adjusted  the  two  wires  separately  in  this  man- 
ner, select  some  well  denned  point  which  the  cross-lines 
are  made  to  bi-sect.  Now  rotate  the  telescope  half  way 


400  A    MANUAL    OF    LAND    SURVEYING. 

round  in  its  wyes.  If  the  point  is  still  bi-sected,  the  tel- 
escope is  collimated.  A  very  excellent  mark  to  use  is  the 
intersection  of  the  cross-lines  of  a  transit  instrument. 

Centre  the  eye-piece  by  the  four  capstan-headed  screws 
nearest  the  eye  end.  This  is  done  by  moving  the  opposite 
screws  in  the  same  direction  until  a  distant  object  under 
observation  is  without  the  appearance  of  a  raise  or  fall 
throughout  an  entire  rotation  of  the  telescope  in  its  wyes. 
The  telescope  is  now  adjusted. 

Next,  bring  the  level  bar  over  two  of  the  leveling 
screws,  focus  the  telescope  upon  some  object  about  300 
:feet  distant,  and  put  on  the  sun-shade.  These  precau- 
tions are  necessary  to  a  nice  adjustment  of  the  level  tube. 
Throw  open  the  two  arms  which  hold  the  telescope  down 
in  its  wyes,  and  carefully  level  the  instrument  over  the 
two  level  screws  parallel  to  the  telescope.  Lift  the  tele- 
scope out  of  its  wyes,  turn  it  end  for  end  and  carefully 
replace  it.  If  the  level  tube  is  adjusted,  the  level  will 
indicate  the  same  reading  as  before.  If  it  does  not,  cor- 
rect half  the  deviation  by  the  two  leveling  screws  and 
the  remainder  by  moving  the  level  tube  vertically  by 
means  of  the  two  cylinder  nuts  which  secure  the  level 
tube  to  the  telescope  tube  at  its  eye-piece  end.  Loosen  the 
upper  nut  with  an  adjusting  pin,  and  then  raise  or  lower 
the  lower  nut  as  the  case  requires,  and  finally  clamp  that 
end  of  the  level  tube  by  bringing  home  the  upper  nut. 
This  adjustment  may  require  several  repetitions  before 
it  is  perfect. 

The  level  is  now  to  be  adjusted  so  that  its  axis  may  be 
parallel  to  the  axis  of  the  telescope.  Rotate  the  telescope 
about  20°  in  its  wyes,  and  note  whether  the  level  bubble 
has  the  same  reading  as  when  the  bubble  was  under  the 
telescope.  If  it  has,  this  adjustment  is  made.  If  it  has 
not  the  same  reading,  move  the  end  of  the  level  tube 
nearest  the  object-glass  in  a  horizontal  direction,  when 
the  telescope  is  in  its  proper  position,  by  means  of  the 
two  small  capstan-headed  screws  which  secure  that  end 
of  the  level  to  the  telescope  tube.  If  the  level  bubble  goes 


LEVELING.  401 

to  the  object-glass  end  when  that  end  is  to  the  engineer's 
right  hand,  upon  rotating  the  telescope  level  toward  him, 
then  these  screws  are  to  be  turned  in  the  direction  of  a 
left-handed  screw,  as  the  engineer  sees  them,  and  vice 
versa.  Having  completed  this  adjustment,  the  level  bar 
itself  must  now  be  made  parallel  to  the  axis  of  the  level. 

To  do  this,  level  the  instrument  carefully  over  two  of 
its  leveling  screws,  the  other  two  being  set  as  nearly  level 
as  may  be;  turn  the  instrument  180°  in  azimuth,  and  if 
the  level  indicates  the  same  inclination,  the  level  bar  is 
adjusted.  If  the  level  bubble  indicates  a  change  of  incli- 
nation of  the  telescope  in  turning  180°,  correct  half  the 
amount  of  the  change  by  the  two  level  screws,  and  the 
remainder  by  the  two  capstan-headed  nuts  at  the  end  of 
the  level  bar,  which  is  to  the  engineer's  left  hand  when 
he  can  read  the  firm's  name.  Turn  both  nuts  in  the  same 
direction,  an  equal  part  of  a  revolution,  starting  that  nut 
first  which  is  in  the  direction  of  the  desired  movement  of 
the  level  bar.  Many  engineers  consider  this  adjustment 
of  little  importance,  preferring  to  bring  the  level  bubble 
in  the  middle  of  its  tube  at  each  sight  by  means  of  the 
leveling  screws  alone,  rather  than  to  give  any  considera- 
tion to  this  adjustment,  should  it  require  to  be  made." 

6.  Leveling  rods  are  made  in  a  variety  of  styles  and 
are  of  two  principal  classes,  viz. :  target  rods  and  speak- 
ing or  self  reading  rods. 

Target  rods  are  made,  of  hard  wood  in  two  or  more 
parts,  which  are  grooved  and  tongued  to  slide  upon  each 
other,  by  which  means  they  are  lengthened  out  to  12  or 
more  feet.  They  are  graduated  to  feet,  tenths  and  hun- 
dredths,  the  decimal  notation  being  more  convenient  for 
computation  than  the  division  into  inches  and  fractions 
of  an  inch.  The  target  is  a  disc  of  brass  made  to  slide 
up  and  down  on  the  rod  and  to  be  clamped  fast  to  the 
rod  at  any  desired  place.  It  is  divided  into  quadrants 
painted  alternately  white  and  red.  When  used  in  level- 
ing the  target  is  moved  up  and  down  on  the  rod  until  the 
horizontal  line  between  these  divisions  is  brought  to  coin- 
cide with  the  line  of  sight  in  the  level.  The  target  has  a 


402 


A    MANUAL    OF    LAND    SURVEYING. 


vernier  attached  by  which  the  distance  on  the  rod  is 
read  to  the  nearest  Tc^7  part  of  a  foot.  In  common 
leveling  it  is  a  useless  refinement  to  carry  the  reading 
to  thousandths  of  a  foot,  as  it  is  out  of  harmony  with 
the  other  conditions  of  the  rod  and  the  work  to  be  done. 
The  target  on  the  rod,  as  a  rule,  is  not  capable  of  being 
set  as  closely  and  accurately  to  the  level  line  as  the  ver- 
nier will  read,  nor  will  the  rod  be  held  so  truly  plumb 
as  to  justify  so  close  a  reading.  Generally  the  line  be- 
tween the  quadrants  of  the  target  is  not  perpendicular 
to  the  rod  and  does  not  coincide  with  the  zero  of  the 
vernier  within  several  thousandths. 

Speaking  rods  are  plain, 
straight  rods,  having  the  gradua- 
tions marked  on  them  so  boldly  and 
distinctly  that  they  can  be  read  from 
the  instrument.  No  targets  are  used 
with  them,  although  some  rods,  like 
the  Philadelphia  rod,  are  made  so  as 
to  be  used  either  as  target  or  speak- 
ing rods.  There  are  many  devices 
for  marking  the  speaking  rod,  all  of 
which  are  intended  to  facilitate  ac- 
curate reading  by  the  observer.  A 
simple  form  of  graduation  and  let- 
tering which  gives  excellent  results 
in  actual  service  is  shown  on  a  re- 
duced scale  in  the  cut.  The  gradu- 
ations are  to  tenths  and  half  tenths 
of  a  foot.  Distances  less  than  half 
a  tenth  are  estimated  by  the  eye. 
This  is  facilitated  by  having  the 
figures  for  tenths  made  either  .04  or 
.06  feet  in  length  and  accurately 
spaced  on  the  rod. 

The  student  having  a  level  and  a 
rod  for  use  in   practice  may  now 
solve  the  following  problems  in  the 
FIG.  78.  fieid: 


I 


LEVELING. 


403 


7.  Prob.  I.  To  find  the  difference  of  level  of  two  points. 
CASE  l.—When  the  difference  of  level  may  be  found  by 
one  setting  of  the  instrument. 


FIG.  7t). 


Suppose  A  and  B  to  be  the  points.  Set  up  the  level 
at  a  point  about  equidistant  from  A  and  B,  though  not 
necessarily  in  a  line  between  them.  Plant  it  firmly  on 
the  ground,  placing  the  legs  so  as  to  bring  the  instrument 
nearly  level,  leaving  as  little  as  possible  to  be  done  with 
the  leveling  screws.  If  set  up  on  yielding  ground  con- 
stant care  will  be  required  to  be  sure  that  the  instrument 
is  level  at  the  instant  the  observation  is  taken.  When 
the  level  is  set  up  on  ice  or  frozen  ground,  the  legs  will 
settle  into  the  frost.  It  is  well  to  set  the  instrument  in 
the  shade  whenever  convenient,  as  the  rays  of  the  sun,  a 
passing  cloud  or  a  sudden  breeze  will  throw  the  instru- 
ment out  of  level  by  causing  unequal  expansion  and  con- 
traction of  the  metal.  In  .precise  leveling  the  instrument 
must  be  shaded.  Having  the  instrument  firmly  planted, 
bring  the  telescope  in  line  with  one  pair  of  the  leveling 
screws  and  turn  them  in  or  out  till  the  bubble  is  brought 
to  the  middle.  Then  bring  the  level  in  line  with  the  other 
pair  and  again  level  it.  Repeat  until  the  bubble  will 
remain  in  the  middle  of  the  tube  through  an  entire  revo- 
lution of  the  telescope  around  the  spindle. 

The  rod-man  holds  the  rod  at  A,  and  its  reading,  Aa 
is  taken.  This  is  called  a  Back  Sight.  All  observa- 
tions on  other  points  taken  at  the  same  setting  of  the 
instrument  are  termed  Fore  Sights.  The  distance  Aa 
shows  how  much  the  line  of  collimation  of  the  level  is 
above  the  point  A  and  is  called  the  height  of  instru- 
ment. The  rod-man  now  holds  the  rod  on  the  point  B 
and  its  reading  is  taken.  The  difference  between  the 


404 


A    MANUAL    OF    LAND    SURVEYING. 


back  sight  and  the  fore  sight  is  the  difference  in  height 
of  the  points  A  and  B.  If  the  back  sight  is  9.20  and  the 
fore  sight  6.40,  then  B  is  2.80  higher  than  A.  If  the  fore 
sight  were  11.45  instead  of  6.40,  then  B  is  2.25  lower  than 
A.  The  rod-man  should  stand  square  behind  his  rod  and 
hold  it  plumb.  Sometimes  small  levels  are  attached  to 
the  rod  to  plumb  it  by.  If  they  are  not  used  the  leveler 
when  necessary  directs  the  rod-man  to  move  the  top  of 
the  rod  to  the  right  or  left  to  plumb  it  that  way,  and  the 
rod-man  also  moves  it  gently  back  and  forth  towards  the 
level,  until  the  smallest  reading  of  the  rod  is  obtained. 
It  is  manifest  that  as  many  points  may  be  taken  as  can 
be  reached  from  the  instrument  and  that  their  relative 
heights  will  be  shown  by  the  distances  they  are  below  the 
horizontal  plane  of  the  instrument,  which  is  told  by  the 
readings  on  the  rod. 

CASE  2.— When  the  difference  of  level  can  not  be  found 
by  one  setting  of  the  instrument. 

Suppose  A  and  E  to  be  the  points,  and  that  it  is 
necessary  to  set  the  instrument  four  times  to  find  the 
difference  between  them.  We  find  by  the  first  setting 
the  difference  between  the  points  A  and  B,  as  already 
described.  We  then  go  forward  and  find  successively  the 
differences  between  the  points  B  and  C,  C  and  D,  and 
D-and  E.  The  algebraic  sum  of  these  differences  is  the 
difference  in  height  of  the  points  A  and  E. 

A  convenient  form  of  field  notes  in  cases  like  the  above 
consists  of  three  columns  as  shown  in  the  following 

Example.—  Required  the  difference  of  level  between 
the  points  A  and  E  from  the  accompanying  notes: 


Sta. 

Back  Sights. 

Fore  Sights. 

A 

3.28 

B 

2.14 

7.15 

C 

3.25 

8.50 

D 

4.70 

3.45 

E 

2.75 

Which  point  is  the  higher,  and  how  much? 


LEVELING.  405 

A  Bench  Mark  or  Bench  is-a  fixed  point  used  for 
reference  in  finding  the  heights  of  other  points.  It  is 
indicated  in  the  notes  by  the  letters  B.  M.  It  is  customary 
to  establish  bench  marks  at  convenient  distances  along 
a  line  of  levels  by  which  the  work  may  be  reviewed,  or  at 
which  it  may  be  resumed  after  temporary  cessation.  The 
most  convenient  permanent  objects  are  selected  for  the 
location  of  these  bench  marks,  such  as  foundation  stones 
in  buildings,  rocks  or  large  boulders,  or  shoulders  cut  in 
the  roots  of  large  tree's,  so  situated  that  the  rod  can  be 
set  up  on  them  and  the  level  readily  taken. 

Where  a  line  of  levels  is  run  taking  a  number  of  points 
it  is  customary  to  refer  the  heights  to  an  assumed  level 
plane  called  a  datum.  This  is  generally  assumed  to  be 
far  enough  below  the  first  or  principal  bench  mark  so 
that  it  shall  be  below  the  lowest  station  likely  to  be  found 
in  any  part  of  the  survey  for  which  it  is  used.  Negative 
heights  are  thus  avoided. 

A  line  of  levels  is  usually  marked  by  stakes  set  at  uni- 
form distances  apart,  marked  and  numbered  consecu- 
tively from  zero  upwards.  100  feet  is  the  distance  most 
usually  adopted  between  stations,  although  in  levels  for 
country  drains  it  is  sometimes  found  more  convenient 
to  space  the  stations  by  chains  to  correspond  with  the 
measures  of  the  land  surveys.  Intermediate  stakes  are 
usually  referred  to  as  plus  stations,  and  are  so  marked 
on  the  stakes  and  in  the  notes.  For  instance,  a  stake  set 
between  stakes  No.  6  and  7  at  40  feet  from  No.  6,  is 
marked  6  -f  40  or  simply  -f  40. 

8.  Prob.  2.  To  find  the  heights  above  a  datum  plane, 
of  several  stations  on  a  given  line. 

•SUGGESTIONS.—  Let  AB  (Fig.  80,  page  406)  be  the 
given  line  and  DP  the  datum  plane  assumed  at  any  con- 
venient distance,  say  10  ft.,  below  a  bench  near  A. 

Set  up  the  level  at  some  convenient  point,  for  example 
between  stations  2  and  3. 

Take  the  reading  of  the  rod  upon  the  bench  and  add  it 


406 


A    MANUAL    OF    LAND    SURVEYING. 


FIG.  80. 


to  the  assumed  height  of  the  bench  above  the  datum. 
The  sum  is  the  height  of  the  instrument. 

Take  the  readings  upon  stations  0,  1,  2,  3,  4  and  5  in 
succession,  and  subtract  each  from  the  height  of  the 
instrument.  The  remainders  are  the  heights,  respect- 
ively, of  those  stations  above  the  datum. 

Carry  the  instrument  forward  to  another  position,  as 
between  stations  6  and  7. 

Take  the  reading  of  the  rod  a  second  time  on  station  5, 
and  add  it  to  the  height  of  station  5  as  before  found. 
The  sum  is  the  new  height  of  instrument,  with  which 
proceed  as  before. 

A  point  used  as  station  5,  as  above  indicated,  is  called 
a  Turning  Point.  In  practice,  a  bench  is  often  adopted 
as  a  turning  point. 

The  reading  of  the  rod  upon  a  turning  point  or  bench- 
mark is  usually  taken  with  somewhat  greater  precision 
than  upon  other  points. 

A  reading  upon  a  bench  or  turning  point  -is  added  to 
the  height  of  the  point  above  the  datum  in  finding  height 
of  instrument;  and  a  reading  upon  any  point  is  sub- 
tracted from  the  height  of  instrument  in  finding  the 
height  of  the  point. 

Accordingly,  an  observation  for  the  former  is  called  a 
Plus  Sight,  denoted  by  -j-S,  and  for  the  latter,  a  Minus 
Sight,  denoted  by  — S. 


LEVELING. 


407 


The  height  of  instrument  is  denoted  by  H.  In.,  and  the 
height  of  any  point  above  the  datum,  by  H.  or  elev. 

The  following  is  an  example  of  the  notes  made  in  solv- 
ing the  above  problem : 


Sta. 

+  s. 

H.  In. 

—  S. 

H. 

Remarks. 

B.M. 

3.426 

13.426 

10.000 

A  stone  20  ft.  S.  E.of  0. 

0 

5.45 

7.976 

I 

7.30 

6.126 

2 

f 

5.35 

8.08 

3 

5.40 

8.03 

4 

6,23 

7.20 

5 

8.274 

3.76 

9.666 

6 

17.040 

5.25 

1  2.69 

7 

5.10 

12.84 

8 

5.00 

3  2.1)4 

9.  Prob.  3.  To  find  the  cut  or  fill,  to  grade,  at  points 
between  two  given  points. 

SUGGESTIONS.—  Let  A  and  B  (Fig.  80)  denote  the 
given  points.  Beginning  at  A,  for  example,  measure  the 
distance  AE,  at  the  same  time  marking  it  off  into  con- 
venient divisions  of  equal  length,  as  33  ft.,  50  ft.,  66  ft., 
or  100  ft.,  for  example,  by  driving  pegs  down  to  the  sur- 
face of  the  ground.  The  last  division  will  ^usually  be  frac- 
tional. Number  the  divisions,  0,  1,  2,  3,  etc.,  beginning 
at  A. 

Find  now  (Prob.  2)  the  heights  of  the  points,  0,  1,  2, 
3,  etc.,  above  some  convenient  datum. 

For  illustration,  suppose  the  heights  to  be  as  given  in 
the  above  Table  (Prob.  2).  Also  suppose  the  height  of 
the  grade  line  at  A  to  be  5  ft.,  and  at  B,  9  ft. 

TJbe  distance  from  A  to  B  consisting  of  8  equal  parts, 
say  of  50  ft.,  we  should  then  have 

(9  ft.  -  5  ft.)  -j-  8  =  0.5  ft.  =-rise  per  station. 


408  A    MANUAL    OF    LAND    SURVEYING. 

Beginning  at  A  or  station  0,  we  have  — 
7.98  _  5.  =  2.98  =  cut  at  0 
6.13  —  5.50  =  0.63  =  cut  at  1 
8.08  —  6.00  =  2.08  =  cut  at  2 
8.03  —  6.50  =  1.53  =  cut  at  3 
6.20  —  7.00  =  —  0.80  =  fill  at  4 
9.67  —  7.50  =  2.17  =  cut  at  5 

etc.        etc.        etc. 

Observe  that  we  take  the  difference  in  height  between 
the  grade*  line  and  the  station  at  each  station ;  and  since 
we  have  here  proceeded  from  lower  to  higher  points  of 
the  grade,  we  have  added  the  rise  of  the  grade  per  station 
to  the  height  of  the  grade  at  the  last  preceding  station. 
Let  the  student  find  the  cut  at  each  station,  beginning 
at  B,  all  other  things  being  as  above. 

Again,  supposing  the  heights  of  the  stations  to  be  as 
above,  let  the  student  find  the  depths  of  cut  and  fill  under 
the  supposition  that  the  height  of  the  grade  at  A  is  6  ft., 
and  at  B,  8.4  ft. 

10.  Drawing  Profile.  Fig.  80  represents  a  section 
formed  by  a  vertical  plane  passing  through  the  points  A 
and  B,  and  meeting  the  datum  plane  in  the  line  DP. 
The  irregular  line  AB  represents  the  intersection  of  the 
vertical  plane  with  the  surface  of  the  ground,  and  is 
called  the  Profile. 

The  manner  of  drawing  the  profile  is  as  follows : 
Draw  a  horizontal  line  to  represent  the  datum  line,  on 
which  lay  off  to  a  convenient  scale  the  distance  between 
the  stations. 

At  the  points  of  division  of  the  datum  line,  erect  per- 
pendiculars, on  which  lay  off  the  surface  heights  of  the 
several  stations,  in  their  order,  but  to  a  scale  usually  ten 
times  greater  than  that  used  for  the  horizontal  distances. 
A  line  drawn  through  the  points  thus  located  forms 
the  profile. 

The  use  of  a  larger  scale  in  drawing  the  vertical  dis- 
tances serves  to  render  the  irregularities  of  the  surface 


DRAINAGE   SURVEYING. 


409 


more  apparent  to  the  eye  than  they  would  be  if  drawn  to 
the  same  scale  with  the  horizontal  measurements. 

The  grade  line  is  drawn  through  any  two  points  at  the 
proper  distances  from  the  datum  line.  The  position  and 
inclination  of  the  grade  line  depend  upon  certain  condi- 
tions required  to  be  fulfilled  by  the  work,  such  as  the 
flowage  of  water,  ease  of  travel,  economy  of  construc- 
tion, etc. 

In  road  work  the  grade  is  often  adopted  with  reference 
to  an  equalization  of  "  cut "  and  "  fill,"  so  that  the  mate- 
rial furnished  by  excavations  shall  make  the  embank- 
ments. The  required  position  of  the  grade  line,  in  order 
to  fulfill  this  condition  most  advantageously,  is  conven- 
iently got  by  stretching  a  thread  across  the  profile,  vary- 
ing the  position  of  the  thread  until  the  areas  intercepted 
by  it  and  the  profile  on  opposite  sides  appear  to  be  equal. 

EXERCISES. 

ii.  1.  Find  depths  of  cut  or  fill,  and  draw  profile  and 
grade  line  from  the  following  notes: 


Sta. 

+s. 

H.  In. 

—  S. 

H. 

H.  Gr. 

Cut. 

Fill. 

0 
1 
2 
3 
3« 

4 
5 
S25 

4.26 
4.12 

14.26 
15.15 

«.30 
8.45 
3.23 
8.20 
4.63 
5.53 
5.75 

10.00 

8.00 
0.575 

Distance  between  stations,  100  ft. 
2—5.  Examples  made  by  the  student  in  the  "  Field." 

II.      DRAINAGE   SURVEYING. 

12.  Of  the  many  applications  of  leveling,  the  most 
common,  perhaps,  in  the  province  of  the  ordinary  sur- 
veyor, is  that  relating  to  drainage.    Almost  every  neigh- 
borhood offers  occasions  for  work  of  this  kind. 

13.  Drains  are  of  two  forms:  the  Open  Drain  or 
Ditch,  and  the  Under  Driin. 

The  former  is  adapted  to  the  case  of  water  lying  upon 


410  A    MANUAL    OF    LAND    SURVEYING. 

the  surface  of  the  ground,  and  the  latter  to  water  under- 
lying the  surface.  Under  drains 'are  usually  discharged 
into  open  drains,  which  are  thus  rendered  an  essential 
auxiliary  to  thorough  drainage. 

14.  Making  the  Survey.— This  will  be,  in  the  first 
place,  a  careful  reconnoissance  of  the  locality  respecting 
the  general  "  lay  of  the  land,"  natural  water  courses,  etc. 
In  this  will  be  determined  the  proper  commencement, 
route  and  terminus  of  the  drain.  The  term  commence- 
ment will  be  here  understood  to  mean  the  upper  end  of 
the  drain,  and  terminus  the  outlet.  The  word  commence- 
ment in  connection  with  open  drains  will  also  be  taken 
as  significant  of  the  proper  place  to  begin  the  survey. 

Preliminaries  having  been  settled,  a  stake  marked  0  is 
driven  at  the  point  of  commencement,  and  the  survey, 
proper,  begins  by  setting  the  transit  over  the  stake  and 
taking  the  bearings  and  distances  of  two  convenient  ob- 
jects near  by  as  witnesses  of  the  point  of  commencement. 
The  location  of  the  commencement  should  be  described 
also  by  distances  and  direction  from  some  neighboring 
monument  or  line  of  original  survey.  Thus,  10  ch.  E. 
and  7.15  ch.  N.  of  V4  post  bet.  Sees.  11  and  14,  T.  2  N.  R. 
5E. 

These  items  are  to  be  entered  in  the  column  of  remarks 
in  the  Transit  book,  opposite  the  station  0. 

The  instrument  is  then  turned  upon  the  first  angle  in 
the  line  of  the  drain  and  its  bearing  entered  in  the  col- 
umn of  bearings  opposite  station  0. 

Ax-men  are  required  in  clearing  away  bushes,  making 
and  driving  stakes,  etc.  Two  chain-men,  the  forward  one 
carrying  a  transit-rod,  now  begin  to  measure  at  0  in  the 
direction  of  the  first  angle,  and  stakes  marked  1,  2,  3,  etc., 
are  driven  at  uniform  distances  from  each  other. 

A  100-ft.  tape  is  a  convenient  measure,  and  locates  the 
stations  at  ordinarily  suitable  distances. 

A  stake  should  be  set  also  at  each  angle  of  the  drain, 
and  its  distance  from  the  last  preceding  station  entered  in 
the  notes.  The  points  of  meeting  of  any  land-lines,  roads, 
etc.,  should  be  noted  by  distances  in  a  similar  manner. 


DRAINAGE   SURVEYING. 


411 


The  number  of  acres  in  farms  whose  lines  are  met  may, 
very  properly,  be  made  a  matter  of  memorandum. 

The  following  is  a  specimen  of  the  form  of  notes  which 
are  taken,  in  accordance  with  the  above  suggestions : 


TRANSIT   NOTES. 


Sta. 

Bearing 

Distance, 
of  Course 

Remarks. 

0 
1 
2 

S.  70°  E. 

0.  A  point  10  ch.  E.  and  7.15  ch. 
N.  of  VA  post  on  line  bet.  Sees. 
11  and  14,  T  2  N.,  K  5  E. 
W.  Oak   15,   N.   23^,°   E.t   57  ft.; 
Hickory  12,  S.  40°   E.,  34  ft. 

3 
4 

M 

Land  owned  by  John  Doe,  80  A.  ; 
about  6  A.   wet. 

5 

" 

5" 

S.  28%°E. 

528  ft. 

5».    1st  Angle. 

G 

•i 

7 

" 

8 

•• 

8« 

<> 

840.    Line  bet.  Sees.  13  and  24. 

9 
10 
11 

K 

B.  Onk  10,  S.  35Vi°  W.,  10  ft.  ; 
W.  Oak  18,  N.  63°  W.,  28  ft. 

Richard   Rowe,    160   A.    on  south, 
30   A.   swamp. 

II80 

East. 

652  ft. 

11"°.    2d  Angle. 

12 

" 

13 

" 

. 

H 

•v'; 

il 

23 

u. 

23« 

M 

1163  ft. 

23".  Terminus    in   drain   by   road 
side  on  Township  line. 

• 

Marked  Boulder,  N.  20°  E.,  15  ft. 
Ash   14,   S.   27°   W.,   10  ft. 

412  A    MANUAL    OF    LAND    SURVEYING. 

15.  Taking  the  Levels.— The  line  of  the  drain  hav- 
ing been  established,  the  next  thing  is  to  take  the  levels. 
This  is  done  in  the  manner  previously  described.    Beside 
the  engineer  or  principal  surveyor,  two  men  are  required 
—  a  rod-man,  and  an  ax-man  to  make  and  drive  pegs. 

The  pegs  should  be  driven  down  even  with  the  surface 
of  the  ground  and  at  such  a  distance  from  the  stakes 
marking  the  stations  that  they  may  be  used  without  dis- 
turbance in  excavating.  Some  practice  driving  them,  say 
six  inches,  in  front  of  the  stakes ;  other  set  them  opposite 
and  at  such  a  uniform  distance  from  the  record  stakes 
as  not  to  be  disturbed  by  the  digging. 

Bench  marks  should  be  made  at  convenient  distances, 
for  example,  at  every  tenth  station,  and  far  enough  from 
the  line  not  to  be  disturbed. 

1 6.  Platting.— The  field  work  having  been  completed, 
the  next  thing  is  to  make  a  plat  of  the  line  and  also  of 
the  sections  or  tracts  of  land  which  will  be  affected  by 
the  drain,  writing  the  owner's  name  and  number  of  acres 
on   each.      On   some   convenient   part   of  the   plat,   the 
courses    and    their    corresponding    distances    should    be 
noted,  also  the  number  of  linear  feet  of  drain  on  each 
separate  tract. 

Next  comes  the  drawing  of  the  profile.  This  is  most 
conveniently  done  by  use  of  paper,  called  Profile  paper, 
prepared  specially  for  the  purpose.  Taking  a  piece  of 
the  proper  width  and  of  sufficient  length  to  contain  also 
the  title  and  necessary  explanatory  notes,  at  the  left 
hand,  we  begin  on  the  edge  next  to  us  and  write  the  num- 
bers of  all  the  stations  in  their  order  toward  the  right, 
upon'  the  vertical  lines.  We  then  mark  with  the  point 
of  a  sharp  pencil  the  point  of  elevation  of  each  station  as 
taken  from  the  column  of  elevations  in  the  level  notes. 
Connecting  the  points  thus  marked,,  by  an  ink  line,  we 
have  the  profile  of  the  surface  of  the  ground  on  the  line 


DRAINAGE    SURVEYING.  413 

of  the  drain.  We  then  take  a  black  thread  and  stretch  it 
on  the  profile  between  the  points  assumed  as  grade,  at  the 
first  and  the  last  station.  From  this  inspection,  it  will  be 
seen  whether  it  is  necessary  or  desirable  to  introduce  one 
or  more  changes  of  grade  between  the  extreme  points  in 
order  to  avoid  objectionable  cuts. 

Having  determined  the  situation  of  the  grade  lines,  we 
then  draw  them  in  their  places,  preferably  with  red  ink. 

Under  the  grade  lines  and  upon  the  vertical  lines  of 
the  several  stations  should  be  written  in  red  ink  the  ele- 
vations of  the  grade,  and  below  that,  in  black  ink,  the 
elevations  of  the  surface.  In  a  similar  manner,  above 
the  profile  may  be  written  first,  in  red  ink,  the  depths  of 
the  cuts,  and,  second,  the  widths  of  the  ditch  at  bottom 
and  top. 

The  names  of  the  land  owners  through  whose  land  the 
ditch  passes,  with  the  number  of  linear  feet  on  each,  may 
be  conveniently  written  upon  the  datum  line. 

17.  The  writer  has  saved  himself  and  assistants  a 
great  many  miles  of  tramping  and  wading  through 
swamps  and  morasses  in  drainage  surveys  by  running 
the  transit  and  level  lines  for  the  drains  both  at  one 
operation.  It  was  found  by  repeated  tests  on  long  lines 
that  the  level  on  the  transit  gave  very  nearly  if  not  quite 
as  accurate  results  in  leveling  as  the  wye  level.  Hence 
the  wye  level  was  left  at  home  and  the  transit  line  and 
levels  were  both  run  at  the  same  time  with  the  transit. 
A  condensed  form  of  keeping  the  notes  was  used.  All 
the  rod  readings  are  kept  in  one  column.  The  back  or 
plus  sights,  to  be  added  to  the  elevation  for  height  of 
instrument,  are  marked  "  B.  S."  The  others  are  all  to  be' 
subtracted  from  "  Ht.  Inst."  for  elevation  of  stations. 
The  following  is  a  sample  extract : 

Commencing  at  a  point  in  the  Section  line  4.53  chains  east 
of  the  quarter  post  between  Sections  11  and  14,  and  running 
thence  S.  16°  W.  Stations  2.00  chains  apart. 


414 


A    MANUAL    OF    LAND    SURVEYING. 


Sta. 

Obs. 

Ht. 

Inst. 

Elev. 

Grade 
Ht. 

Cut 

Remarks. 

B.S.on 
B.M. 

4.96 

104.96 

100.00 

On     Elm     40'     to 
rt.   of   Sta.   1. 

0 
1 

5.21 
5.30 

99.75 
99.66 

96.00 
95.90 

3.75 
3.76 

Elm     and     Black 
Ash. 

2 
3 

5.28 
5.46 

99.68 
99.50 

95.80 
95.70 

3.88 
3.80 

+50,    enter    thick 
Willows. 

4 

5.72 

99.24 

95.60 

3.64 

5 

5.83 

99.13 

95.50 

3.63 

-f  GO       Angle  rt.  12°  24'=  S.  28°  24'  W.    Cross  line  fence  be- 
tween Smith  and  Jones. 

C 

5.84 

99.12 

B.S. 

2.91 

102.03 

6 

2.95 

99.08 

95.40 

3.68 

Open  marsh.    Saw 

grass. 

7 

3.06 

98.97 

95.30 

3.67 

1 8.  Depth  and  Width.— The  depth  of  a  drain  obvi- 
ously depends  upon  the  situation  of  the  grade  line  with 
respect  to  the  surface.  In  adjusting  the  grade  line  it  is 
more  important  to  guard  against  the  drain  being  too 
shallow  rather  than  too  deep;  most  open  drains  are  too 
shallow. 

Again,  it  should  be  taken  into  account,  if  the  drain  is 
to  run  through  soft  marshes  and  hard  ridges,  that  the 
soft  ground,  on  the  withdrawal  of  the  water,  will  settle; 
ard  so  the  drain  may  need  to  be  dug  deeper  in  some 
places  than  would  otherwise  be  necessary. 

The  necessary  width  of  a  drain  of  given  depth  and 
grade  depends  upon  the  quantity  of  water  it  is  required 
to  discharge  in  a  given  time. 


The  width  at  the  top  is  determined  from  the  width  at 
the  bottom  and  the  slope  or  inclination  given  the  sides, 


DRAINAGE   SURVEYING.  415 

which  is  usually  from  one  to  one  and  one-half  feet  on  the 
horizontal  to  each  foot  in  depth. 

19.  Quantity  of  Discharge.—  The  amount  of  water 
which  a  drain  may  discharge  in  a  given  time  obviously 
depends  upon  the  area  of  the  water-way  or  cross-section 
of  the  drain  and  the  velocity  of  the  stream. 

Thus,  denoting  by  Q  the  quantity  of  discharge,  by  a 
the  area  of  the  water-way,  and  by  v  the  mean  velocity  of 
discharge,  we  should  have 

Q=av  (1) 

As  an  approximate  formula  for  computing  the  mean 
velocity  of  water  flowing  in  an  open  canal  of  uniform 
cross-section  and  fall,  Trautwine  gives  the  formula 


•i 


a/*  X  8975     )  % 
-   I  -.1089   (2) 


in  which  V=  mean  velocity  in  feet  per  second,  a  =  area 
of  water-way  in  square  feet,  f=fall  in  feet  per  foot,  and 
p  —  wet  perimeter  or  the  water  border  of  the  channel. 

REMARK. —  In  applying  the  above  formula,  It  is  customary  to 
use  9000  for  8975  and  .11  for  .1089. 

Example.—  Required  the  velocity  and  the  capacity  of  a 
drain  5  ft.  wide  at  the  bottom,  the  sides  having  a  slope  of 
1  to  1,  depth  of  water  3  ft.,  and  the  fall  2  ft.  to  1,000  ft. 

Solution.^- Width  at  top  =  5  ft. +  2  X  3  ft.  =  11  ft. 

Area  of  water-way  =  1%  (11  ft.  +  5  f t. )  =  24  sq.  ft. 

Wet  perimeter  =  5  ft.  -f  6\/2  ft.  =  13.5  ft. 

Fall  per  foot  =  0.002  ft. 

(  24X0.002X9000  \  % 

Substituting  in  (2),7=<(  V  —0.11 

(  13.5  ) 

=5.55. 

Substituting  in  (1),  #  =  24X5.55  =  133.2  cu.  ft 
per  second,  or  11,508,480  cu.  ft.  per  day. 


416  A    MANUAL    OF    LAND    SURVEYING. 

Trautwine  gives  also  the  following  formula,  with  the 
remark  that  it  is  applicable  also  to  sewers : 


in  which  a  and  p  are  as  above  described,  and  F  is  the  fall 
in  feet  per  mile. 

REMARK. —  In  connection  with  the  above  formulas,  as  well  as 
with  others  of  similar  import,  Trautwine  re'peats  again  and 
again  the  caution  that  they  are  to  be  regarded  only  as  approx- 
imately true. 

Table  XII  shows  approximately  the  number  of  acres 
served  by  drains  having  bottom  widths  of  1  to  10  ft.,  with 
side  slopes  of  1  to  1,  and  various  rates  of  fall  per  station, 
on  the  supposition  of  1  inch  rainfall  in  24  hours,  one- 
half  of  which  reaches  the  drain. 

20.  Amount  of  Rainfall. —  All  calculations  of 
requisite  capacity  of  drains  must  be  based  upon  the 
probable  amount  or  number  of  inches  of  rainfall  in  a 
given  time.  The  soil,  however,  acts  as  a  reservoir  up 
to  the  point  of  saturation,  depending  upon  its  texture, 
keeping  from  the  drains  altogether  a  portion  of  the 
rainfall,  which  passes  off  by  evaporation  or  is  absorbed 
by  plants. 

The  average  annual  rainfall  in  Michigan,  Indiana, 
Illinois  and  Missouri  is  about  35  inches.  In  Ohio,  for  a 
period  of  ten  years,  it  was  reported  to  be  37.86  inches. 

In  the  matter  of  rainfall  in  Michigan,  we  are  indebted 
to  Prof.  Carpenter  for  the  following  data : 

"  By  a  consultation  of  the  meteorological  records  of  the  Agri- 
cultural College  we  learn  that,  although  large  showers  in  which 
the  rainfall  exceeds  one  inch  occur  comparatively  seldom  (on  the 
average  only  four  times  a  year),  yet  they  bring  with  them  twen- 
ty-eight per  cent  of  our  total  rainfall  during  that  period,  and 
consequently  they  must  be  fully  provided  for  in  any  works  for 
thorough  drainage.  The  following  table  is  compiled  from  the 
meteorological  records  kept  at  the  college,  and  shows  the  com- 
parative depth  and  number  of  showers  from  the  months  of  March 
to  December  for  five  years.  The  last  column  shows  the  total  per- 
centage of  rainfall  in  all  the  showers  of  a  given  depth.  The  last 


DRAINAGE   SURVEYING. 


417 


column  but  one  shows  the  total  percentage  of  the  number  of 
showers  compared  with  the  whole  number.  Although  this  table 
is  not  extended  sufficiently  far  back  to  give  very  accurate  results, 
it  is  thought  (since  one  year's  rainfall  does  not  differ  greatly 
from  that  of  another  year)  to  be  sufficiently  reliable  to  produce 
data  for  any  ordinary  case  of  farm  drainage  in  this  part  of  the 
United  States. 

TABLE   OF   SHOWERS   FROM   MARCH   TO   DECEMBER. 


Depth  of  Rain- 
fall in  Inches. 

Number  of  Showers. 

Percentage  of  Total. 

1872 

1873 

1874 

1875 

187C 

Total 

No.  of 
Showers. 

Am't  of 
Rainfall. 

.00  to    .25  . 

.25  to     .50  
.50  to     .75  
.75  to  1.00  
1.00  to  1.25  
1.25  to  1.50  
1.50  to  1.75  
1.75  to  2.00  
2.00  to  2.25  
2.25  to  2  50 

19 

20 
C 
2 

40 
14 
8 
G 

28 
13 
6 
5 

33 
9 
10 
2 

43 
11 
5 
3 
2 
3 

105 
67 
35 
18 
2 
8 
3 
2 

o~ 

54.2 
22.0 
11.5 
OC.O 
00.7 
02.C 
01.0 
00.7 

OOJ3 
00.3 

~~66.Y~ 

17 
21 
21 
13 

2 
9 

4 
3 

2 
2 
.... 

3 
1 

2 
-- 



„ 
1 

2.50  to  2.75  
2.75  to  3.00  
3.00  to  3.25  



"-- 

-- 

1 

Totals  











304 

100.00 

100 

"The  amount  of  discharge  of  drains  as  compared  with  the 
rainfall  is  usually  estimated  at  about  50  per  cent.  So  that  in 
order  to  produce  thorough  drainage  it  is  necessary  to  assume 
that  the  capacity  of  the  drains  shall  be  sufficient  to  carry  off 
during  twenty-four  hours  one-half  the  water  that  fell  the  pre- 
vious twenty-four  hours.  The  probability  of  the  rainfall  in  any 
day  exceeding  one  inch  is  so  slight  that  we  shall  be  safe  in 
assuming  as  the  necessary  carrying  capacity  of  drains  one-half 
of  3,630  cu.  ft.,  or  1,815  cu.  ft.  of  water  for  each  acre  drained." 

21.  Under  Drains  are  formed  in  various  ways; 
sometimes  of  brush,  rails  or  loose  stone  trenched  in, 
sometimes  of  tubes  made  of  logs  or  of  iron,  sometimes 
of  plank  or  of  brick  or  stone  laid  in  cement,  and  again  of 
earthen  tubes,  of  which  there  are  various  forms,  called 
Tiles. 

The  prevailing  method  of  under-drainage  for  agricul- 
tural purposes  consists  in  the  use  of  cylindrical  tiles, 
which  are  made  of  different  sizes  and  usually  about  a 
foot  in  length. 


418  A    MANUAL    OF    LAND    SURVEYING. 

It  is  of  this  form  of  under  drain,  only,  that  we  propose 
to  write  briefly. 

22.  Surveying  for    Under   Drains.— Very   much 
of  what  has  been  said  upon  surveying  for  the  ditch  or 
open  drain  applies  also  to  the  tile  drain.    The  same  pre- 
liminary  inspection   is  required   to   determine   the   best 
location  of  the  outlet  and  the  proper  directions  of  trunk 
and  branch  lines.     Indications  as  to  source  of  water, 
whether  from  springs  on  the  premises  or  on  lands  sit- 
uated above,  whether  from  rainfall,  merely,  upon  the 
particular  tract  or  also  as  flowing  off  from  neighboring 
areas;  the  directions  of  slopes,  whether  of  surface  or  of 
underlying  strata;  the  character  of  the  soil,  etc.,  all  have 
to  be  carefully  observed  and  their  bearing  duly  con- 
sidered. 

23.  Location  of  Drains.— As  above  intimated,  any 
well-conducted  survey  for  under  drains  contemplates  the 
execution  of  a  system  of  drains  working  together  and 
depending  upon  each  other.     This  will  include  usually  a 
principal  drain,  called  a  Main,  and  lateral  drains,  called 
Minors,  which  discharge  into  the  main.    In  ah  extended 
system,    auxiliary    mains    called    Sub-Mains    are    also 
introduced. 

Since  it  is  the  direct  office  of  the  minors  to  remove  the 
surplus  water  from  the  ground,  it  is  of  the  first  impor- 
tance that  they  be  so  located  as  successfully  to  perform 
their  functions.  To  do  this  requires  the  exercise  of  care- 
ful judgment  on  the  part  of  the  engineer,  respecting  the 
proper  directions  of  the  minors  and  also  their  distances 
from  each  other.  Equal  care  is  requisite  also  in  regard 
to  the  location  of  the  main,  so  as  properly  to  receive  the 
water  from  the  minors  and  discharge  it  at  the  principal 
outlet. 

As  a  rule,  the  main  should  be  located  at  the  foot  of  the 
regular  slopes,  or  along  the  valleys  of  the  field;  and,  in 


DRAINAGE   SURVEYING.  419 

general,  the  minors  should  run  directly  down  the  slopes, 
discharging  themselves  obliquely  into  the  main. 

Cases,  however,  will  sometimes  occur  that  require  de- 
parture from  the  above  rules,  but  these  are  to  be  regarded 
as  "  exceptions  which  prove  the  rule." 

The  distances  of  the  minors  from  each  other  will  be 
governed  largely  by  the  character  of  the  soil  as  to  per- 
meability, and  to  some  extent  by  the  depth  of  the  drains. 
In  a  porous  soil,  as  a  general  rule,  the  deeper  the  drain 
the  further  it  will  draw. 

Circumstances  are  infinitely  varied.  Every  situation  is 
a  new  one  and  must  be  treated  on  its  own  merits.  None 
but  the  most  general  instruction  on  this  point  can  be 
given  In  any  treatise.  About  as  practical  a  suggestion 
as  may  be  afforded  the  student  is,  Go  into  the  field  and 
there  mix  plenty  of  brains  with  your  work. 

24.  Running  the  Lines. — Having  settled  the  ques- 
tion of  the  proper  system  of  drains  to  be  adopted,  the 
next  thing  to  be  done  is  to  lay  out  and  measure  the  lines. 
This  is  perhaps  most  conveniently  done  in  the  case  of 
under  drains,  by  beginning  at  the  outlet,  measuring  and 
staking  out,  first,  the  main  lines  of  the  system  and  then 
the  branches. 

A  distance  of  50  ft.  between  stations  is  a  convenient 
one  in  tile  draining.  In  some  instances,  as  where  the  fall 
is  very  slight,  a  less  distance  may  be  desirable ;  in  others 
a  -greater  one  may  give  equally  good  results.  In  addition 
to  the  stakes  driven  at  the  uniform  distances  of  the  sta- 
tions, a  stake  should  mark  the  entrance  of  each  minor, 
and  the  distance  to  it  should  be  entered  in  the  notes,  in 
the  usual  manner.  Such  stakes  mark  the  points  of 
beginning  in  running  out  the  minors. 

To  facilitate  examinations  for  "  faults,"  the  points  of 
entrance  of  the  branches  in  the  main  drain  should  be 
established  by  witnesses. 


420  A    MANUAL    OF    LAND    SURVEYING. 

25.  Taking  the  Levels. — This  is  done  in  the  same 
manner  as  in  the  case  of  open  drains,  but,  perhaps,  with 
a  somewhat  greater  degree  of  care  and  precision.  The 
point  assumed  for  the  outlet  must,  of  course,  be  suf- 
ficiently low  to  receive  all  the  water  of  the  field;  and  at 
the  same  time  the  outlet  ought  to  be  high  enough  to  be 
at  all  times  above  the  back  water  of  the  stream  into 
which  the  drain  empties.  A  drain  is  of  little  more  use 
under  a  violation  of  the  latter  condition  than  under  a 
disregard  of  the  former. 

In  assuming  the  grade,  due  consideration  must  be  had 
for  proper  depth  consistently  with  required  fall. 

The  depth  of  an  under  drain  should  be,  at  the  least, 
two  feet;  all  the  better  if  three  or  four  feet  in  most  soils. 

Henry  F.  French,  author  of  "  Farm  Drainage,"  says : 
"  We  cannot,  however,  against  the  overwhelming  weight 
of  authority,  and  against  the  reasons  for  deeper  drain- 
age, which  to  us  seem  so  satisfactory,  conclude  that  even 
three  feet  is,  in  general,  deep  enough  for  under  drains. 
Three-foot  drains  will  produce  striking  results  on  almost 
any  wet  lands,  but  four-foot  drains  will  be  more  secure 
and  durable,  will  give  wider  feeding-ground  to  the  roots, 
better  filter  percolating  water,  warm  and  dry  the  land 
earlier  in  Spring,  furnish  a  larger  reservoir  for  heavy 
rains,  and,  indeed,  more  effectually  perform  every  office 
of  drains." 

Accordingly,  the  rule  should  be  to  approximate  as 
closely  as  possible  to  what  are  thus  regarded  as  desirable 
depths,  admitting  depths  very  much  below  the  standard 
only  when  we  must,  in  order  to  have  any  drains  at  all. 

Upon  the  question  of  necessary  amount  of  fall,  with 
which  the  surveyor  is  so  often  confronted  in  connection 
with  the  requirement  of  desirable  depths,  it  is  to  be  ob 
served  in  the  first  place  that  large,  deep  streams  require 
less  fall  than  small  ones;  and,  again,  the  form  and  the 


DRAINAGE   SURVEYING.  421 

condition  of  the  channel  have  much  to  do  with  the  move- 
ment of  water. 

"  It  has  been  found  in  practice  that  a  water-course 
thirty  feet  wide  and  six  feet  deep  will  flow  at  the  rate 
of  one  mile  per  hour,  with  a  fall  of  no  more  than  six 
inches  per  mile." 

Examples  are  cited  of  successful  operation  of  drains 
with  three  inches  or  even  two  and  one-half  inches  fall  to 
one  hundred  feet,  or  even  on  a  dead  level. 

The  contour  of  the  ground  will  determine  the  grade 
of  the  drain,  which  will  be  given  all  the  fall  there  is. 
A  level  drain  will  work  successfully  provided  it  is  laid 
to  a  true  line  and  kept  free  from  obstructions.  The 
writer  successfully  lowered  two  lakes,  covering  about 
120  acres,  by  an  open  ditch  two  feet  wide  on  the  bottom, 
a  mile  long,  and  laid  perfectly  level.  The  less  fall  there 
is,  the  larger  should  be  the  tile  used  and  the  greater 
should  be  the  care  taken  in  laying  them  to  a  true  grade 
line  and  keeping  out  leaves  or  other  obstructions. 

Changes  of  grade,  though  undesirable,  are  admissible 
when  not  easily  avoided.  If  possible,  the  heaviest  grades 
should  be  in  the  direction  of  the  outlet.  When  this  can 
not  be,  it  may  be  desirable  to  introduce  silt  wells  at  points 
of  any  considerable  change  of  grade. 

The  heights  of  the  outlets  of  minor  drains  into  the 
main  are  usually  the  heights  of  grade  in  the  main  drain 
for  the  same  points. 

26.  Constructing  the  -Drain. — The  principal  point 
is  the  method  of  opening  the  trench  and  laying  the  tiles 
on  the  grade  line. 

To  do  this  systematically  requires  a  measuring  rod  six 
or  eight  feet  in  length  divided  into  feet,  tenths,  and  hun- 
dredths  of  feet,  the  larger  divisions  being  numbered  up- 
ward, as  in  the  ordinary  leveling  rod.  A  cord  or  wire. 


422  A    MANUAL    OF    LAND    SURVEYING. 

also  is  needed,  which  is  to  be  stretched  above  the  line  of 
the  drain  and  adjusted  to  a  position  parallel,  to  the  grade 
line.  This  is  done  by  inverting  the  measuring  rod  on  the 
grade  peg  and  bringing  the  cord  or  wire  to  the  division 
of  the  rod  indicating  the  cut  at  that  point.  The  cord  is 
thus  placed  at  the  full  length  of  the  measuring  rod  from 
the  grade  line  or  intended  bottom  of  the  trench. 

The  cord  may  be  held  each  fifty  or  one  hundred  feet  by 
two  slats,  each  about  seven  feet  long,  and  movable  about 
a  bolt  passing  through  a  little  distance  from  the  upper 
end.  These  are  called  Shears.  The  cord  or  wire  is  pre- 
vented from  slipping  by  a  couple  of  turns,  and  is  tied  to 
a  stake  eight  or  ten  feet  from  the  shears. 

Another  device  consists  in  the  use  of  stakes  or  posts 
driven  on  opposite  sides  of  the  ditch,  and  connected  with 
a  cross-bar  arranged  so  that  either  end  may  be  raised  or 
lowered  to  a  level,  and  fastened  to  the  posts  by  a  clamp 
and  thumb-screw.  The  cross-bars  being  adjusted  to  the 
proper  height,  as  above  described,  the  cord  or  wire  is 
drawn  tightly  across  them,  directly  over  the  center  line 
of  the  drain. 

Again,  single  stakes  or  posts,  driven  on  one  side  of  the 
ditch,  each  having  attached  at  right  angles  an  arm  which 
may  be  raised  or  lowered,  and  secured  in  place  by  a  clamp 
and  screw,  are  sometimes  employed. 

By  such  means  as  the  above,  the  ditch  is  readily  dug  to 
just  the  proper  depth,  and  the  tile  laid  to  grade  with  ex- 
ceeding accuracy  and  with  great  rapidity.  The  proper 
distance  from  the  top  of  the  tile  to  the  cord  may  be  indi- 
cated by  an  arm  attached  to  the  measuring  rod. 

27.  Size  of  Tile. — The  size  of  tile  required  in  a  given 
case  will  depend  upon  the  quantity  of  water  to  be  re- 
moved and  the  fall  available  to  remove  it.  Formulas 
are  given  in  works  upon  hydraulics,  to  express  the  veloci- 
ty and  discharge  of  water  flowing  in  pipes,  but  the  condi- 


DRAINAGE    SURVEYING.  "  423 

tions  are  so  different  in  case  of  tiles  that  such  formulas, 
at  best,  give  only  the  most  roughly  approximate  results. 

Thus,  for  example,  the  following,  which  is  Poncelet's 
formula  : 


F=48 


L  +  54D 


in  which,  F=  approximate  velocity  in  feet  per  second, 
D  =  diameter  of  pipe  in  feet,  H  =  total  head  in  feet,  and 
L  =  total  length  of  pipe  in  feet. 

Having  found  the  velocity,  we  have 

Discharge  in  cu.  ft.  =  vel.  X  cross-section  of  pipe. 

Tables  XII  and  XIII  are  used  for  the  above  purpose, 
the  latter  quite  extensively  by  drainage  engineers  and  has 
been  found  to  give  good  results. 

As  regards  size  of  tile  for  main  and  sub-main  drains  a 
good  authority  says,  "  that  can  be  regulated  only  by  the 
person  in  charge  of  the  drainage  at  any  particular  place, 
after  seeing  the  land  opened  up  and  the  minor  drains  dis- 
charging. As  a  rule,  a  circular  pipe  of  three  inches  inter- 
nal diameter  will  discharge  the  ordinary  drainage  of  six 
statute  acres,  and  give  sufficient  space  for  the  circulation 
of  the  air." 

This  estimate  is  based  upon  an  amount  of  annual  rain- 
fall of  from  twenty-six  to  thirty  inches,  which  differs  but 
slightly  from  -that  of  Michigan  and  adjoining  States. 

In  addition  to  the  above,  it  may  be  remarked  that  if  the 
fall  in  the  main  is  slight,  a  larger  size  of  tile  would  be 
required  than  if  the  fall  was  considerable. 

And,  again  in  order  to  provide  suitably  for  the  accu- 
mulation of  water  which  occurs  toward  the  outlet,  a 
larger  size  may  be  there  required  than  that  used  in  the 
upper  part  of  the  main. 


424  A    MANUAL    OF    LAND    SURVEYING. 

28.  Protection  at  Outlets. — The  outlets  of  under- 
drains  should  be  protected  by  some  construction  to  pre- 
vent the  earth  from  falling  down  in  front  of  the  drain. 
A  retaining  wall  of  masonry  laid  in  hydraulic  cement  is 
the  best  provision  for  the  purpose.    The  outlets  should  be 
protected  also  by  a  coarse  grating  of  some  sort  in  front 
of  the  tile  to  prevent  muskrats  and  other  creatures  from 
getting  in. 

A  common  practice  is  to  introduce  at  the  outlet  a  box 
made  of  plank  a  few  feet  in  length,  into  which  the  tile  is 
made  to  discharge. 

29.  Silt  Well. — This  is  a  well  sunk  below  the  level 
of  the  tile  for  catching  the  silt  gathered  by  the  drains 
above  it.    It  serves  also  the  purpose  of  affording  a  means 
of  inspecting  the  working  of  the  drains.     Silt  wells  may 
be  constructed  with  a  view,  chiefly,  to  facilitating  the 
movement  of  the  water  at  an  abrupt  bend  in  the  dram. 
And  again,  they  may  be  constructed  somewhat  with  ref- 
erence to  convenience  of  obtaining  a  pail  of  water  for 
any  purpose,  in  the  field. 


SURVEYORS'  TABLES. 


SUGGESTIONS  ON  USE  OF  TABLES. 


TABLES. 


SUGGESTIONS   TO  YOUNG  SURVEYORS   ON    THE   USES   OF 
THE   TABLES. 

Traverse  Table.  — The  table  calculated  to  quarter 
degrees  is  adapted  to  the  simplest  work  of  compass 
surveying,  where  great  accuracy  is  neither  required 
nor  expected.  When  the  transit  is  used,  and  the  angles 
are  taken  to  minutes  or  less,  the  author  prefers  the  tables 
of  logarithms  and  logarithmic  sines  and  cosines  to  any 
traverse  table  yet  made.  They  are  capable  of  any  re- 
quired degree  of  accuracy,  and  require  the  use  of  no  more 
figures  than  the  ordinary  traverse  table.  In  transit  work, 
where  latitudes  and  departures  are  to  be  calculated,  it  is 
well  to  refer  the  angles  of  all  lines  to  a  common  base, 
just  as  in  compass  surveying  all  lines  are  referred  to  the 
meridian  as  a  base.  Then,  in  any  course, 

Latitude  =  co-sine  of  angle  X  length  of  the  course. 

Departure  =  sine  of  angle  X  length  of  the  course. 
Using  the  logarithmic  tables,  this  is  a  short  and  simple 
computation. 

Example  1.— Angle,  36°  22'.  Distance,  47.63.  Eequired 
the  latitude  and  departure. 

Log.  of  47.63  =  1.677881  to  which  add 
log.  sine,  36°  22'  =  9.773018 

11.450899  the  log  of  28.244-  =  departure. 

Log.  of  47.63  =  1.677881  to  which  add 
log.  cos.,  36°  22'  =  9.905925 

11.583806  the  log.  of  38.35+  =  latitude. 

2.  Course  N.  57°  21'  20"  E.  34.36^  chains.  Required 
the  latitude  and  departure. 


A  MANUAL  OF  LAND   SURVEYING. 


1.  The   Table   of    Tangents  is   convenient    in 
estimating  courses  of  lines  to  be  run. 

Example  1. — From  the  quarter  post  on  the  east  side  of 
Section  2  I  wish  to  run  a  line  for  a  road  straight  to  a 
point  80  rods  north  of  the  southwest  corner  of  Section  30. 
What  course  shall  I  run  ? 

Solution.— Distance  west,  5  miles;  distance  south,  4.25 
miles,  which  divided  by  5  equals  the  natural  tangent  of 
the  angle  which  the  course  makes  with  an  east  and  west 
line,  =  .850.  Find  this  number  in  the  table  of  natural 
tangents  and  take  out  the  corresponding  angle,  =  40°  22', 
which  is  the  same  as  S.  49°  38'  W. 

2.  What  is  the  course  from  the  village  of  Climax,  at 
the  east  quarter  post  of  Section  3,  Township  3  south, 
Range  9  west,  to  the  village  of  Richland,  at  the  southwest 
corner  of  Section  14,  Township  1  south,  Range  10  west? 
To  the  village  of  Schoolcraft,  at  the  southeast  corner 
of  Section  19,  T.  4  S.,  R.  11  W.,  from  Climax?    What  to 
Schoolcraft  from  Richland? 

2.  The  Table  of  Secants  is  convenient  for  finding 
the  hypothenuse  of  a  triangle,  thus  simplifying  many 
computations  in  the  field.  Secants  not  given  in  the  table 
may  be  found  by  interpolation  or  by  the  formula: 

1 

Secant  = . 

cosine 

The  following  example  indicates  one  of  the  practical 
applications  in  the  field: 

Example.— Lots  in  a 
city  are  laid  out  with 
their  lines  perpendicu- 
lar to  N  Street  and 
running  through  to  M 
Street.  Required  the 
width  (x)  of  the  lots  on 
M  Street. 

Call  the  width  of  the 

N.5t lots    on    N   Street  r. 

Measure  the  angle  A. 


FIG.  8i. 


SUGGESTIONS  ON  USE  OF  TABLES. 


Ill 


Then  x  =  r,  sec.  A.  If  r  =  100,  as  is  common,  x  may  be 
taken  directly  from  the  table.  If  r  =  100,  A  *=  21°  4(X, 
then  re  =  107.6.  In  laying,  out  such  lots  it  is  generally 
easier  and  quicker  to  measure  this  distance  on  the  street 
line  than  it  is  to  set  up  the  transit  for  each  lot  line  and 
run  it  in. 

3.  Table  of  Departures.— This  table  has  many 
convenient  uses,  of  which  a  few  examples  ar,e  given. 

Examples. — 1.  I  wish  to  stake  out  a  line  along  an  oid 
hedge  row  from  quarter-post  to  section  corner.  On  one 
side  is  a  clear  field.  I  go  to  the  section  corner,  and  make 
an  offset  of  25  links  and  set  up  a  flag.  1  then  go  to  the 
quarter-post,  and,  making  an  equal  offset,  find  that  I 
cannot  see  the  flag;  so  I  offset  until  I  can  see  it— say  37 
links  more.  I  sight  to  the  flag,  find  from  the  table  of 
departures  the  angle  corresponding  to  37  links  at  a  dis- 
tance of  40  chains  =  32'.  turn  off  the  angle  on  the  transit, 
and  run  the  line  back  parallel  with  the  section  line, 
setting  stakes  on  the  true  line,  by  62  link  offsets,  as 
often  as  required. 

2.  To  run  a  true  half-quarter-line  when  one  end  is 
inaccessible. 

Fig.  82  repre- 
sents the  whole 
section,  and  ab  the 
line  to  be  run. 

Bisect  eg,  setting 
stake  at  a.  Meas- 
ure the  angle  acd, 
which  we  will  call 
89°  24'.  By  the 
field  notes  the 
north  line  of  the 
section  measures 
80.22,  hence  <zc=» 
20.05J.  The  south 
line  measures 


FIG.  82. 


IV 


A  MANUAL  OF  LAND    SURVEYING. 


79.63,  one-fourth  of  which  is  19.90|.  Hence  the  section 
line  and  half -quarter-line  converge  at  the  rate  of  20.055  — 
19.9075  =  .1475  chains  per  mile.  From  the  table  of 
departures  we  find  the  corresponding  angle  to  be  a  little 
more  than  6'.  Hence  we  make  the  angle  gab  6'  greater 
than  acd  =  89°  30-K,  and  run  the  line  accordingly. 

The  foregoing  are  given  as  samples  of  many  labor- 
saving  uses  of  the  tables,  which  the  young  surveyor 
should  study  out  and  be  prompt  to  avail  himself  of 
when  the  occasion  requires. 

TRIGONOMETRIC  FUNCTIONS  AND  FORMULAE. 


C 

FIG.  83. 

Then  sin  A  =BC 
tan  A  =  DF 
sec  A  =  AD 
versin  A  =  CF 
exsec  A  =  BD 
chord  A  =  BF 


7  G  Let  Fig.  83  represent 
the  various  trigonomet- 
ric functions.  Let  ABC 
represent  the  angles,  and 
dbc  the  sides  opposite  in 
the  right  triangle  formed 
by  the  radius,  sine  and 

,  cosine.  Other  parts  as 
shown  in  the  figure. 

cos  A  =AC 
cot  A  =  HG 
cosec  A  =  AG 
coversin  A  =  BK 
coexsec  A  —  BG 
chord  2  A  =  2BC. 


Tables  of  these  functions  are  calculated  with  radius 
AH=±\. 


Sin  4      =  —  =C9SJ?          cosJ.         =  —  =  sin.B 
c  c 

a  6 

Tan  A   -=  —  =cot.B          cot4         =  —  =  tan.B 
6  a 


FORMULAS.  V 

C  C 

Sec  A     =  —  =  cosec  B  cosec  A      =  —  =secB 
b  a 

a — 6  c — a 

Vers  A  = f=-  covers  B  coversin  A  = = vers  B 

c  c 

c — 6  c — a- 

Exsec^l== =  coexsec.B  coexsec  A  — =exsecJi? 

6  a 


CcsinA  =  b  tan  A  fccos  A  =  a  cot  A 

a=iccosB  =  6  cot  B         b  =    c  sin  B^  a  tan  B 


a 


sin  A      cos  A 
a  b 


cos  B      sin  2? 


— a} 


(7  =  90°  = 

ab 

Area  =  — . 
2 


cot  ^1  —  cot  5 

Useful  in  measur- 
ing heights  o  f 
objects  or  passing 
obstacles  in  line. 


FIG.  84. 


VI  A  MANUAL  OF  LAND  SURVEYING. 

SOLUTION  OF  OBLIQUE  TRIANGLES. 

Let  ABC  represent  the  angles,  and  abc  the  opposite 
sides,  of  any  oblique  triangle.       . 


Given. 


A,B,a 


A,a,b 


C,  a,  b 


AtB,C,a 


Sought. 


B,C, 


Area 
A 


Area 
Area 


180°  —  (A  +  B).     b=—. sinB. 


A  +  B). 

>.     C=  180—  (A  +  B). 


—  —  - 
sin  A 


tan  i  (A—B)  =     T    tan 


Area  =  K=  \  ab  sin  C. 
f< 

S) 


then  sin  i^=J^-6)^-°). 
be 


tan 


/(s  —  6)  (s  —  c 
\      *  (.v  —  d) 


sin  J.  =r-  vxs  (s — a)  (s  —  6)  (s  —  c). 
2  (.«  —  b}  (s  —  c). 

versin  A  = 
Area  = 


be 


Area  = 


(s  —  a)  (s—b)  (s  —  c). 
a2  sin  B  sin  C 


TABLE  I.      LOGARITHMS  OF  NUMBERS. 


TABLES. 

LOGAEITHMS    OF    NTJMBEKS 

FROM 

1    TO    10000. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0000000 

26 

414973 

51 

1  707570 

76 

1880814 

2 

0  301030 

27 

431364 

52 

1716003 

77 

1  886491 

3 

0  477121 

28 

447158 

53 

1  724276 

78 

1892095 

4 

06020t>0 

29 

462398 

54 

1  732394 

79 

1  897627 

5 

0698970 

30 

477121 

55 

1740363 

80 

1  903090 

6 

0  778151 

31 

491362 

56 

1  748188 

81 

1908485 

7 

0845098 

32 

505150 

57 

1755875 

82 

1  913814 

8 

0903090 

33 

518514 

58 

1763428 

83 

1  919078 

9 

09M243 

34 

531479 

59 

1770852 

84 

1  924279 

10 

1  000000 

35 

544068 

60 

1  778151 

85 

1929419 

11 

041393 

36 

556303 

61 

1  785330 

86 

1  934498 

12 

079181 

37 

568202 

62 

1792392 

87 

1  939519 

13 

113943 

38 

579784 

63 

1  799341 

88 

1  944483 

14 

146128 

39 

591065 

64 

1  806180 

89 

1949390 

15 

176091 

40 

602060 

65 

1  812913 

90 

1954243 

16 

20*120 

41 

612784 

66 

1  819544 

91 

1959041 

17 

230449 

42 

623249 

67 

1  826075 

92 

1  963788 

18 

255273 

43 

633468 

68 

1  832509 

93 

1  968483 

19 

278754 

44 

643453 

69 

1  838849 

94 

1  973128 

20 

301030 

45 

653213 

70 

1  845098 

95 

1  977724 

21 

322219 

46 

662758 

71 

1  851258 

96 

1982271 

22 

342423 

47 

672098 

72 

1  857332 

97 

1  986778 

23 

361728 

48 

681241 

73 

1  863323 

98 

1  991238 

24 

380211 

49 

690196 

74 

1  869232 

99 

1  995686 

25 

1  397940 

50 

698970 

75 

1  87o061 

100 

2000000 

TABLE   I.      LOGARITHMS  OF  NUMBERS. 


No. 

o 

I 

2 

8 

4  |   5 

6 

7 

s 

9 

Diff. 

100 

000000 

000434 

000868 

001301 

001734 

002166  002598  003ft29 

003461 

003891 

432 

1 

4321 

4751 

5181 

5609 

6038 

6466  1  6894  1  7321 

7748 

8174 

428 

2 

8600 

9026 

9451 

9876 

010300 

010724 

011147011570 

011993 

012415 

424 

3 

012837 

013259 

013680 

014100 

4521 

4940 

5360 

5779 

6197 

6616 

'419 

4 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

020361 

020775 

416 

5 

021189 

021603 

022016 

022428 

022841 

023252 

023664  1024075 

4486 

4896 

412 

6 

5306 

5715 

6125 

6533 

6942 

7350 

77571  8164 

8571 

8978 

408 

7 

9384 

9789 

030195 

030600 

031004 

031408 

031812 

032216 

032619 

033021 

404 

8 

033424 

033826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

9 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

04C207 

040602 

040998 

396 

110 

041393 

041787 

042182 

042576 

042969 

043362 

043755 

044148,044540 

044932 

393 

1 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053   8442 

8830 

389 

2 

9218 

9606 

9993 

050380 

050766 

051153 

051538 

051924  052309 

052694 

386 

3 

053078 

053463 

053846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

382 

4 

6905 

7286  7666 

8046 

8426 

8805 

9185 

9563 

9942 

060320 

379 

5 

060698 

061075  061452 

061829 

062206 

062582 

062958 

063333 

063709 

4083 

376 

6 

4458 

48321  52061  5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557]  8928  9298 

9668 

070038 

070407 

070776 

071145 

071514 

369 

8 

071882 

072250 

072617  072985 

073352 

3718 

4085 

4451 

4816 

5182 

366 

9 

5547 

5912 

6276,  6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

079543  079904 

080266 

080626 

080987 

081347 

081707 

082067 

082426 

360 

1 

082785 

083144 

083503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

2 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

3 

9905 

090258 

090611 

090963 

091315 

091667 

092018 

092370 

092721 

093071 

351 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

100026 

346 

6 

100371 

100715 

101059 

101403 

101747 

102091 

102434 

102777 

103119 

3462 

343 

7 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

110253 

338 

9 

110590 

110926 

111263 

111599 

111934 

112270 

112605 

112940 

113275 

3609 

335 

130 

113943 

114277 

114611 

114944 

115278 

115611 

115943 

116276 

116608 

116940 

333 

1 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

120245 

330 

2 

120574 

120903 

121231 

121560 

121888 

122216 

122544 

122871 

123198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

b!31 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

130012 

323 

5130334 

130655 

130977 

131298 

131619 

131939 

132260 

132580 

132900 

3219 

321 

6 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

7 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

315 

8 

9879 

140194 

140508 

140822 

141136 

141450 

141763 

142076 

142389 

142702 

314 

9143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140  146128 

146438 

146748 

147058 

147367 

147676 

147985 

148294 

148603 

148911 

S09 

1 

9219 

9527 

9835 

150142 

150449 

150756 

151063 

151370 

151676 

151982 

307 

2 

152288 

152594 

152900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

4 

8362 

8664 

8965 

9266 

9567 

9868 

160168 

160469 

160769 

161068 

301 

5 

161368 

161667 

161967 

162266 

162564 

162863 

3161 

3460 

3758 

4055 

299 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

8 

170262 

170555 

170848 

171141 

171434 

171726 

172019 

172311 

172603 

172895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

176381 

176670 

176959 

177248 

177536 

177825 

178113 

178401 

178689 

289 

1 

8977 

9264 

9552 

9839 

180126 

180413 

180699 

180986 

181272 

181558 

287 

2 

181844 

182129 

182415 

182700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

190051 

281 

5 

190332 

190612 

190892 

191171 

191451 

191730 

192010 

192289 

192567 

2846 

279 

6 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

7 

5900 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

8 

8657 

8932 

9206 

9481 

9755 

200029 

200303 

200577 

200850 

201124 

274 

9 

201397 

201670 

201943 

202216 

202488 

2761 

3033 

3305 

3577 

3848 

272 

No. 

O 

1 

'2  I   » 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE   I.      LOGARITHMS  OF  NUMBERS. 


No 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

Diff. 

16020412C 

204391  2046G32049& 

205204 

205475  205746  206016  206286 

20655G 

271 

1 

6826 

7096  j  7365!  7634 

7904 

8173  8441   8710!  8979  9247 

269 

2 

9515 

9783.210051  210319  210586 

210853  211121 

211388-211654  211921 

267 

3212188 

212454 

2720!  298G 

3252 

3518!  3783  4049!  4314 

4579 

266 

4 

4844 

5109 

5373  5638 

5902 

G1GG  6430  66941  6957 

7221 

264 

I 

7484 

7747 

8010  8273 

8536 

8798  9060  9323J  9585i  9846 

262 

6220108 

220370220631220892 

221153 

221414  221675  221936|222196;222456 

261 

71  2716 

2976  3236  3496 

OK 

40151  4274 

45331  4792|  5051 

259 

8  6309 

5568|  5S26I  6084 

6342 

6600!  6858 

7115   7372!  7630 

258 

9  7887 

8144 

8400  8657 

8913 

9170  9426 

9682   9938230193 

256 

170230449 

230704  230960!231215 

231470 

231724'231979 

2322341232488  232742 

264 

1 

2996 

3250 

3504|  3757 

4011 

4264   4517 

4770 

5023  5276 

253 

2 

5528 

5781 

6033!  6285 

6537 

6789  |  7041 

7292 

7544 

7795 

252 

3  8046 

8297 

85481  8799 

9049 

9299  9550 

9800  240050  240300 

250 

4240549 

240799^241048  241297 

241546 

241795242044 

242293!  2541  |  2790 

249 

6 

CC38 

3286  3534 

3782 

4030 

4277 

4525 

4772 

5019!  6266 

248 

G 

5513 

6759  6006 

6252 

64991  6745 

6991 

7237 

7482|  7728 

246 

7 

7973 

8219  8464 

8709 

8954  9198  9443 

9687 

9932250176 

245 

1 

250420250664:250908 

251151 

251395  251638  251881  252125  252368i  2610 

243 

9 

2853 

3096,  3338 

3580 

3822 

4064 

4306   4548   4790   5031 

242 

180'255273 

255514  2H755 

255996 

256237  256477  '256718!  256958]257198  257439 

241 

l|  7679 

7918 

8158 

8398 

8637  1  8377  9116;  9355   9594   9833 

239 

2,260071 

260310 

260548 

260787 

2G1025  261263  261501  261739  261976  262214 

238 

3  2451 

2688 

2925 

3162  3399 

3636 

3873 

4109   4346   4582 

237 

4  4818 

6054 

5290 

5525  5761 

5996 

6232 

6467   6702;  6937J  236 

5  7172 

7406 

7641 

7875^  811w  8344 

8578 

88121  9046!  9279 

234 

6 

9513 

9746  9980 

270213  27014G  270679 

270912 

271144271377271609 

233 

7 

271842  272074  272306 

25381  2770  3001 

3233 

34641  3696;  39271  232 

8 

4158  4389 

4620 

4850J  6081 

5311 

5542 

6772   6002 

6232|  230 

9 

6462  66S2 

6921 

7151   7380 

7609 

7838 

8067   8296 

8525j  229 

190 

278754'278982  279211 

279439  2796G7  279895 

280123 

280351280578 

280806  228 

1 

281033281261,281488 

281715i281942  282169 

2396 

2622  j  2849 

3075  227 

2 

3301 

3527 

3753 

3979  4205 

4431 

4656 

4882  5107 

5332,  226 

3 

6557 

6782 

6007 

6232  6456 

6S81 

6906 

7130 

7354 

7578  225 

4 

7802 

8026 

8249 

8473  8696 

8920 

9143 

9366 

9589 

9812  223 

6290035 

290257 

290480 

290702290925 

291147 

291369 

291591  291813 

292034.  222 

61  2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804!  4025 

4240J  221 

7  4466 

4687 

4907 

6127 

5347 

6567 

5787 

6007J  6226 

6440 

220 

8  6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198   8416 

8635 

21  n 

9  8853 

9071 

9289 

9507 

9725 

9943300161 

300378300595 

3008131  218 

2CO  301030 

301247 

301464 

301681 

301898 

302114  302331 

302547302764 

302980  217 

1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706  4921 

6136 

216 

2  5351 

6566 

5781 

5996 

6211 

6425 

6639 

6854  7068 

7282 

214 

3  7496 

7710 

7924 

8137 

8351 

8564 

8778 

8S91 

9204 

9417 

213 

4 
B 

9630 
311754 

9843  310056 
311966  2177 

310268 
2389 

310481 
2600 

310693 
2812 

310906 
3023 

311118 
3234 

311330 
3445 

311542|  212 
3656  211 

1 

3867 

4078  4289 

4499 

4710 

4920 

5130 

6340 

5551 

5760  210 

7 

5970 

6180  6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854  209 

8 

8063 

8272  8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938:  208 

I 

320146 

320354J320562  320769 

320977 

321184 

321391 

321598 

321805 

322012 

207 

216 

322219 

322426322633322839 

323046 

323252 

323458 

323665 

323871 

324077 

206 

1 

4282 

4488 

4694 

4899 

5105 

6310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

330008 

330211 

203 

4 

330414 

330617  330819 

331022 

331225 

331427 

331630 

331832 

2034 

2236 

202 

5 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4263 

202 

I 

4454 

4655 

4856 

5057 

5257 

6458 

6658 

5859 

6059 

6260 

201 

1 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

8 

8456 

8G56 

8855 

9054 

9253 

9451 

9650 

9849 

340047 

340246 

199 

9 

340444  340642  340841 

341039 

341237 

341435 

341632 

341830 

2028 

2225 

198 

No- 

0 

1  |  2 

3 

4 

5 

6 

7 

8 

9 

Dlff. 

TABLE  L      IX)GAEITHMS  OF  NUMBERS. 


0. 

o 

1 

2 

a. 

4 

5 

6 

7 

§ 

0 

ff. 

220 

342423 

342620 

(42317 

343014 

(43212 

(43409 

(43606 

3802 

(43999 

344196 

97 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

96 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

95 

3 

8305 

8500 

8694 

8880 

9083 

9278 

9472 

9666 

9860 

350054 

194 

4 

350248 

350442 

{50636 

350829 

351023 

(51216 

351410 

(51603 

(51796 

1989 

193 

6 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

6834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

8 
9 

7935 
9835 

8125 
360025 

8316 
360215 

8506 
360404 

8696 
360593 

8886 
360783 

9076 
360972 

9266 
361161 

9456 
361350 

9646 
361539 

190 
189 

230 

361728 

361917 

362105 

362294 

362482 

362671 

362859 

363048 

363236 

363424 

188 

1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

6675 

6862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

70143 

70328 

370513 

70698 

70883 

185 

5 

371068 

71253 

371437371622 

371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

4748 

4932 

6115 

6298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 
1 

380211 
2017 

380392 
2197 

380573 
2377 

380754 
2557 

380934 
2737 

381115 
2917 

381296 
3097 

381476 
3277 

381656 
3456 

381837 
3636 

181 
180 

c 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5240 

6428 

179 

* 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

i 

5 

7390 
9166 

7568 
9343 

7746 
9520 

7923 
9698 

8101 
9875 

8279 
90051 

8456 

390228 

8634 
90405 

8811 
390582 

8989 
390759 

178 
177 

( 

390935 

391112 

391288 

391464 

391641 

1817 

1993 

2169 

2345 

2521 

176 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

392 

4101 

4277 

176 

j 

4452 

4627 

4802 

4977 

6152 

6326 

6501 

567 

5850 

6025 

175 

9 

6199 

6374 

6548 

6722 

6896 

707 

7245 

741 

7592 

7766 

174 

25( 

397940 
9674 

98114 

9847 

398287 
400020 

398461 
400192 

398634 
40036 

39880 
40053 

398981 
40071 

399154 

40088 

399328 
401056 

39950 
40122 

173 

17J 

i 

401401 

401573 

1745 

1917 

208 

226 

2433 

260 

2777 

294 

172 

3121 

3292 

3464 

3635 

380 

397 

414 

432 

4492 

466 

171 

i 

483- 

6005 

5176 

6346 

651 

668 

685 

602 

6199 

637 

171 

J 

6540 

6710 

6881 

705 

722 

739 

756 

773 

790 

807 

170 

8240 

8410 

8579 

874 

891 

9087 

925 

942 

959 

9764 

169 

1 

9933 
411620 

410102 

1788 

410271 
1956 

410440 
2124 

410609 
229 

41077 
246 

41094 
2629 

41111 

2796 

411283 
2964 

41145 
313 

169 
168 

j 

3300 

3467 

3635 

380 

397 

413 

4305 

447 

463 

4806 

167 

260 

414973 

415140 

415307 

41547 

41564 

415808 

41597 

41614 

41630 

41647 

167 

664 

6807 

6973 

713 

7306 

747 

763 

7804 

797 

8135 

166 

2 
3 

830 
995 

8467 
42012 

8633 

420286 

879 
42045 

8964 
42061 

912S 

42078 

929 

42094 

9460 
42111 

962c 
42127 

979 
42143 

165 
165 

4 

421604 

176 

1933 

209 

226 

242 

2590 

2754 

291 

308 

164 

£ 

3246 

3410 

3574 

373 

390 

406 

4228 

4392 

455 

471 

164 

I 

488 

5045 

5208 

537 

5534 

669 

6860 

602, 

618 

634 

163 

651 

667 

6836 

6999 

716 

732 

7486 

764 

781 

797 

162 

813 

829 

MS 

862 

8783 

8944 

9106 

926? 

942 

959 

162 

9 

975 

991 

430075 

430236 

43Q39 

43055 

430720 

43088 

43104. 

43120 

161 

270 

] 

431364 
296 

43152 
3130 

43168E 
329C 

43184 
345C 

432007 
361 

43216 
377 

432328 
3930 

43248 

409C 

43264 
424 

32808 
440 

161 
160 

j 

456 

4729 

4888 

604 

620 

636 

652 

668E 

584 

600 

159 

j 

61& 

6322 

648 

664 

679 

695 

711 

727* 

»  743. 

759 

159 

^ 

775 

790 

806 

822 

838 

854 

870 

8851 

>  901 

917 

158 

i 

933 

949 

964 

9806 

9964 

44012 

44027 

44043r 

44059^ 

44075 

158 

( 

44090S 

441066 

44122 

44138 

44153 

169 

185 

2001 

)  216 

232 

157 

• 

248 

263 

279 

295C 

310( 

326, 

341 

357( 

J  373 

388 

157 

i 

404 

420 

435 

451 

466 

482 

498 

613' 

r  529 

64*! 

156 

i 

5604 

676C 

591 

607 

622 

638 

653 

6692 

1  6841 

700. 

155 

Mo 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dlff. 

TABLE   I.      LOGARITHMS  OF  NUMBERS. 


No, 

o 

1 

2 

3 

Dlff. 

280 

447158,447313 

447468 

447623 

447778 

447933 

448088 

448242 

448397 

448552 

155 

1 

8706J  8861 

9015 

9170 

9324  9478 

9633 

9787)  9941 

450095 

154 

2 

450249;450403 

450557 

450711 

450865,451018 

451172 

451  326!  451479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

48451  4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366;  6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

7 

78821  8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

8 

9392  9543 

9694 

9845 

999£ 

460146 

460296 

460447 

460597 

460748 

151 

9 

460898  ;  461048 

461198 

461348 

461499 

1649 

1799 

1948  2098 

2248 

150 

290 

462398 

462548 

462697 

462847 

462997 

463146 

463296 

463445463594 

463744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

470116 

470263 

470410 

470557 

470704 

470851 

470998 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

£610 

147 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

40/1 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

6526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

•  6832 

6976 

145 

300 
1 

477121 
8566 

477266 
8711 

477411 
8855 

477555 
8999 

477700 
9143 

477844 
9287 

477989 
9431 

478135 
9575 

478278 
9719 

4784221  145 
986C1  144 

2 

480007 

480151 

480294 

480438 

480682 

480725 

480869 

481012 

481156 

481290  144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731  143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157  143 

5 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579  142 

6 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997!  142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410!  141 

8 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818  141 

9 

9958 

490099 

490239 

490380 

490520 

490661  490801 

490941 

491081 

491222  140 

310 

491362 

491502 

491642 

491782 

491922 

492062492201 

492341 

492481 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

6 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

6 

9687  9824 

9962 

500099 

500236 

500374 

500511 

500648 

500785 

500922 

137 

7 

501059;501196 

501333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

505150 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

9471 

9606 

9740 

9874 

510009 

510143 

510277 

510411 

134 

4 

510545 

510679 

510813 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

6 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

8 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

518514 

518646 

518777 

518909 

519040 

519171 

519303 

519434 

519566 

519697 

131 

1 

9828 

9959 

520090 

520221 

520353 

520484 

520615 

520745 

520876 

521007 

131 

2 

521138 

521269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

3 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

5 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

7 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

8 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

530072 

128 

9 

530200 

530328 

530456 

530584 

530712 

530840 

530968 

531096 

531223 

1351 

128 

No. 

O 

1 

a 

3 

4 

£ 

6 

7 

8 

9 

Dlff. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


No. 

O 

1 

2 

o 

4    5 

6 

7 

8 

9 

Dlff. 

340 

531479 

531607 

531734531862 

531990 

532117 

532245 

532372532500 

532627 

128 

1 

2754 

2882 

30091  3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

2 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

6041 

5167 

127 

3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

5 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

125 

7 

540329 

540455 

540580 

540705 

540830 

540955 

541080 

541205 

1330 

1454 

125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544068 

544192 

544316 

544440 

544564 

544688 

544812 

544936 

545060 

545183 

124 

1 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

2 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

550106 

123 

5 
6 

550228 
1450 

550351 
1572 

550473 
1694 

550595 
1816 

550717 
1938 

550840 
2060 

550962 
2181 

551084 
2303 

551206 
2425 

1328 

2547 

122 
122 

7 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

3GO 

556303 

556423 

556544 

556664 

556785 

556905 

557026 

557146 

557267 

557387 

120 

1 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120. 

2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

3 

9907 

560026 

560146 

560265 

560385 

560504 

560624 

560743 

560863 

560982 

120 

4 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

'1936 

2055 

2174 

119 

5 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

8 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 
1 

568202 
9374 

568319 
9491 

568436  568554 
9608  9725 

568671 
9842 

568788 
9959 

568905  569023 
570076  570193 

569140 
570309 

569257 
570426 

117 
117 

2 

570543 

570660 

570776  570893 

571010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

6 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

9 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

579898 

580012 

580126 

580241 

580355 

580469 

580583 

580697 

580811 

114 

1 

580925 

581039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

4 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

5 

5461 

5574 

1  5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

8 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

.  9615 

9726 

9838 

112 

9 

9950 

590061 

590173 

590284 

590396 

590507 

590619 

590730 

590842 

590953 

112 

390 

591065 

591176 

591287 

591399 

591510 

591621 

591732 

591843 

591955 

592066 

111 

1 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

2 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

j 

5496 

5606 

5,717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

8 

9883 

9992 

600101 

600210 

600319 

600428 

600537 

600646 

600755 

600864 

109 

9 

600973 

601082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE   I.      LOGARITHMS  OF   NUMBERS. 


No. 

O 

1 

2 

3 

4 

5 

6 

7    8 

O 

Diff. 

400 

602060 

602169 

602277 

602386 

602494 

602603 

602711  602819  602928 

603036 

108 

1 

3144 

3253 

3361 

3469 

3577 

3686 

3794 

3902  4010 

4118 

108 

2 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

108 

3 

5305 

5413 

6521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133  7241 

7348 

107 

5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

82051  8312 

8419 

107 

6 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274!  9381 

9488 

107 

7 

9594 

9701 

9808 

9914610021610128 

610234  610341  610447 

610554 

107 

8 

610660 

610767 

610873 

610979   1086!  1192 

1298   1405 

1511 

1617 

106 

9 

1723 

1829 

1936 

2042 

2148  2254 

2360  2466 

2572 

2678 

106 

410 

612784 

612890 

612996 

613102 

613207  613313 

613419 

613525 

613630 

613736 

106 

1 

3842 

3947 

4053 

4159 

426>  4370 

4475 

4581 

4686 

4792 

106 

2 

4897 

5003 

5108 

5213 

5319  5424 

5529 

5634 

5740 

5845 

105 

3 

5950 

6055 

6160 

6265 

6370  6476 

6581 

6686 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

74201  7525 

7629 

7734 

7839 

7943 

105 

K 

o 

8048 

8153 

8257 

8362 

8466  8571 

8676 

8780 

8884 

8989 

105 

6 

9093  9198 

93021  9K)6  9511;  9615 

9719 

9824 

9928 

620032 

104 

7 

620136620240 

620344  620448  620552  620656 

620760 

620864 

620968 

1072 

104 

8 

1176!  1280 

1384 

1438 

1592   1695 

1799 

1903 

2007 

2110 

104 

9 

2214  2318 

2421 

2525 

2628  2732 

2835 

2939 

3042 

3146 

104 

420 

623249623353 

623456  623559  623663  623766 

623869 

623973 

624076 

624179 

103 

1 

4282 

4385 

4488j  4591  |  4695  4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

6518  6621 

5724  5827 

5929 

6032 

6135 

6238 

103 

3 

6340 

6443 

6546  6648  6751  1  6853 

6956 

7058 

7161 

7263 

103 

4 

7366 

7468 

7571   76731  7775!  7878 

7980 

8082 

8185 

8287 

102 

5 

8389 

8491 

8593  8695 

8797  8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613!  9715 

9817  9919 

080081 

630123630224 

63<>:»; 

102 

7630428 

630530  630631  630733  630835  63093(3 

1038 

11391  1241 

1342 

102 

8 

1444 

1545 

1647i  1748 

1849  1951 

2052 

2153  2255 

2356 

101 

9 

2457 

2559 

2660  2761 

2862J  2963 

3064 

3165  3266 

3367 

101 

430 

633468 

633559 

633670  633771 

633872633973 

634074 

634175634276 

634376 

101 

1 

4477 

4578 

4679  4779 

4880;  4981 

5081 

6182  6283 

5383 

101 

2 

5484 

6584 

6685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

3 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189  7290 

7390 

100 

4 

7490 

7590 

7690 

7790  7890 

7990 

8090 

8190  8290 

8389 

100 

5 

8489 

8589 

86S9 

8789  8888 

8988 

9088 

9188  9287 

9387 

99 

6 

9486 

9586 

9686  9785  9885  9984640084 

640183640283 

640382 

99 

7640481 

640581  640680  640779  640879  640978 

1077 

1177 

1276 

1375 

99 

8 

1474 

1573,  1672   1771   1871 

1970  2069 

2168 

2267 

2366 

99 

9 

2465 

2563  2662  2761  2860 

2959  3058 

3156  3255 

3354 

99 

440 

643453 

643551  643650  643749  643847 

643946  644044  644143  644242 

644340 

98 

1 

4439 

4537  4636  4734 

4832 

4931  6029  5127 

5226 

5324 

98 

2 

5422 

5521 

5619 

5717 

5815 

5913  6011 

6110 

6208 

6306 

98 

3 

6404 

6502 

6600 

6698 

6796 

G894  6992 

7089 

7187 

7285 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872!  7969  8067 

8165!  8262 

98 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945  9043 

9140  9237 

97 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919650016 

650113650210 

97 

7650308 

650405  650.302  650399  650696,650793 

650890  0987 

1084   1181 

97 

8 

1278 

1375  Ii72  1569  1666  1762 

1859 

1956 

2053:  2150 

97 

9 

2246 

2343  2140  2536  2633 

2730 

2826 

2923 

3019 

3116 

97 

450653213 

653309  653405  653502  653598  653695 

653791 

653888 

653984 

654080 

96 

1 

4177 

4273  4369  4465  4562  4658 

4754 

4850 

4946 

6042 

96 

2 

5138 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

96 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

5 

8011 

8107  8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

6 

8965  9060  9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

95 

7 

9916  660011  660106 

660201 

660296 

660391 

660486 

660581 

660676 

660771 

95 

8 

660865  0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

95 

9 

1813)  1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

95 

No. 

0 

123 

4 

5 

G 

7    8 

9 

Diff. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


No.   O 

1 

3 

3 

4 

5 

6 

7 

8 

9 

iff. 

460  662758 
1  3701 
2  4642 
3  5581 
4  6518 
5  7453 
6  8386 
7  9317 
8  670246 
9  1173 

62852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 
70339 
1265 

2947 
3889 
4830 
5769 
6705 
7640 
8572 
9503 
0431 
1358 

63041 
3983 
4924 
5862 
6799 
7733 
8665 
9596 
70524 
1451 

63135 
4078 
5018 
5956 
6892 
7826 
8759 
9689 
0617 
1543 

63230 
4172 
5112 
6050 
6986 
7920 
8852 
9782 
0710 
1636 

63324 
4266 
5206 
6143 
7079 
8013 
8945 
9875 
0802 
1728 

63418 
4360 
5299 
6237 
7173 
8106 
9038 
9967 
0895 
1821 

4454 
5393 
6331 
7266 
8199 
9131 
0060 
0988 
1913 

63607 
4548 
5487 
6434 
7360 
8293 
9224 
70153 
1080 
2005 

94 
94 
94 
94 
94 
93 
93 
93 
93 
93 

470  672098 
1  3021 
2  3942 
3  4861 
4  5778 
5  6694 
6  7607 
7  8518 
8  9428 
9680336 

72190 
3113 
4034 
4953 
5870 
6785 
7698 
8609 
9519 
80426 

2283 
3205 
4126 
5045 
5962 
6876 
7789 
8700 
9610 
580517 

72375 
3297 
4218 
5137 
6053 
6968 
7881 
8791 
9700 
680607 

72467 
3390 
4310 
5228 
6145 
7059 
7972 
8882 
9791 
80698 

72560 
3482 
4402 
5320 
6236 
7151 
8063 
8973 
9882 
80789 

72652 
3574 
4494 
5412 
6328 
7242 
8154 
9064 
9973 
80879 

2744 
3666 
4586 
5503 
6419 
7333 
8245 
9155 
80063 
0970 

2836 
3758 
4677 
5595 
6511 
7424 
8336 
9246 
80154 
1060 

572929 
3850 
4769 
5687 
6602 
7516 
8427 
9337 
680245 
1151 

92 
92 
92 
92 
92 
91 
91 
91 
91 
91 

480  81241 
1  2145 
2  3047 
3  3947 
4  4845 
6  5742 
6  6636 
7  7529 
8  8420 
9  9309 

81332 
2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 

81422 
2326 
3227 
4127 
5025 
5921 
6815 
7707 
8598 
9486 

681513 
2416 
3317 
4217 
5114 
6010 
6904 
7796 
8687 
9575 

j81603 
2506 
3407 
4307 
5204 
6100 
6994 
7886 
8776 
9664 

581693 
2596 
3497 
4396 
5294 
6189 
7083 
7975 
8865 
9753 

E581784 
2686 
3587 
4486 
5383 
6279 
7172 
8064 
8953 
9841 

81874 
2777 
3677 
4576 
5473 
6368 
7261 
8153 
9042 
9930 

81964 
2867 
3767 
4666 
5563 
6458 
7351 
8242 
9131 
90019 

682055 
2957 
3857 
4756 
5652 
6547 
7440 
8331 
9220 
690107 

90 
90 
90 
90 
90 
89 
89 
89 
89 
89 

490  690196 
1   1081 
2  1965 
3  284 
4  372 
5  460 
6  548 
7  635 
8  722 
9  810 

690285 
1170 
2053 
2935 
381 
469 
556 
6444 
731 
818 

90373 
125 
214 
302 
390 
478 
565 
653 
7404 
827 

69046 
134 
223 
311 
399 
486 
5744 
66l 
749 
836 

90550 
1435 
2318 
319 
407 
495 
583 
6706 
757 
844 

90639 
1524 
2406 
3287 
4166 
5044 
591 
679 
7665 
853 

90728 
1612 
2494 
3375 
4254 
513 
600 
688 
775 
862 

90816 
1700 
2583 
3463 
434 
521 
6094 
696 
783 
8709 

9090 
178 
2671 
3551 
4430 
5307 
6182 
7055 
7926 
-8796 

690993 
1877 
275 
363 
451 
539 
626 
714 
801 
888 

89 

88 
88 
88 
88 
88 
87 
87 
87 
87 

500  69897 
1  9838 

69905 
992 

69914 
70001 

69923 
70009 

69931 
700184 

699404 

70027 

69949 
70035 

69957 
700444 

699664 
700531 

69975 
70061 

87 

87 

270070 
3  156 
4  2431 

700790 
1654 
251 

087 
174 
2602 

096 
1827 
268 

1050 
191 
277 

113 
1999 
286 

1222 
208 
294 

1309 
217 
303 

1395 
2258 
31  1£ 

148 
2344 
320 

86 
86 
86 

5  3291 
6  4151 
7  5008 
8  5864 

337 
4236 
5094 
594 

346J 
4325 
617< 
603f 

354 
440 
»  526v 
>  6126 

3635 
449 
535C 
6206 

372 

457 
6436 
629 

380 
4665 
5522 
637 

389 
475 
560 
646 

397£ 
483'J 
569C 
654' 

406^ 
492 
577 
66 

86 
86 
85 

9  671* 

(  680 

688* 

5  697 

705 

714 

722 

731 

746X 

74 

85 

510  707576 
1  8421 
2  9276 
3  71011' 

)  70765 
8506 
)  935 

r  71020* 

j  104 

707746 
859 
944< 
71028' 
113' 

)  70782 
L  867 
)  952 
r7103 
I  12 

70791 
876 
96CK 
71045< 
130 

707996 
884 
9694 
71054 
138 

70808 
893 
977 
71062 
147 

708166 
901 
986k 
71071 
1554 

708251 
916X 
9945 
71079^ 
163< 

708336 
91 
7100 
08 
17 

85 
85 
85 

85 
84 

6  180' 
6  26M 

r  189 

\    ty-r'. 

197 
281 

5  206 
5  290 

21 
29 

222 
30 

23 
31 

239 
32 

248 
332 

256< 
34 

84 

84 

7  349 
8  433 
9  516 

L  35 
)  44 

r  52 

365 
449 
533 

)  37 

r  45 

5  54 

38 
466 
5501 

39 

47 
55 

39 
48 
56 

40 
49 
§7 

416 
56X> 
583 

42 
50 
59 

84 
84 

Mo   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dlff. 

520 

716003 

716087 

716170 

716254 

716337 

716421 

716504 

716588 

716671 

716754 

83 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

83 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

720077 

83 

5 

720159 

720242 

720325 

720407 

720490  720573  720655  720738  720821 

0903 

83 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

82 

g 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 
1 

724276 
5095 

724358 
5176 

724440 

5258 

724522 
5340 

724604 

5422 

724685 
6503 

724767 
5585 

724849 
5667 

724931 
5748 

725013 
6830 

82 

82 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

82 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

6 
7 

9165 
9974 

9246 
730055 

9327 
730136 

9408 
730217 

9489 

730298 

9570 
730378 

9651 
730459 

9732 
730540 

9813 
730621 

9893 
730702 

81 
81 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 
1 

732394 
3197 

732474 
3278 

732555 
3358 

732635 
3438 

732715 
3518 

732796 
3598 

732876 
3679 

732956 
3759 

733037 
3839 

733117 
3919 

80 
80 

2 

3999 

4079 

4160 

4240 

4320 

4400 

"4480 

4560 

*  4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

6359 

6439 

6519 

80 

4 

6599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

80 

5 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

7 

7987 

80G7 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

g 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

9 

9572 

9651 

9731 

9810 

9889 

9968 

740047  740126 

740205 

740284 

79 

550 

740363 

740442 

740521 

740600 

740678 

740757 

740836  740915 

740994 

741073 

79 

1 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

-4215 

78 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

6 

5075 

5153 

5231 

6309 

6387 

5465 

6543 

6621 

6699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6666 

78 

g 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748188 

748266 

748343 

748421 

748498 

748576 

748653 

748731 

748808 

748885 

77 

8963 

9040 

9118 

9195 

9272 

93501  9427 

9504 

95821  9659 

77 

2 

9736 

9814 

9891 

9968 

750045;750123 

750200 

750277 

760354 

75O431 

77 

3 

750508 

750586 

750663 

750740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

'4 

1279 

1356 

1483 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

•77 

5 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2686 

2663 

2740 

77 

6 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

g 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

6036 

76 

9 

6112 

5189 

5265 

6341 

6417 

6494 

6570 

5646 

5722 

6799 

76 

570 
1 

755875 
6636 

755951 
6712 

756027 
6788 

756103 
6864 

756180  756256 
6940  7016 

756332 
7092 

756408 
7168 

756484 
7244 

756560 
7320 

76 
76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

76 

3 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9692 

76 

5 

9668 

9743 

9819  9894 

9970760045 

760121 

760196  760272 

760347 

76 

6 

760422 

760498 

760573760649 

760724 

0799 

0875 

0950|  1025 

1101 

76 

7 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

76 

g 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

9 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3353 

75 

No. 

0 

1 

2  !  3 

4 

5 

6 

7 

8 

9 

Dlff. 

10 


TABLE  I.      LOGARITHMS  OF  NUMBERS. 


Ho.]-  0 

1 

2 

3 

4 

5 

6. 

7 

8 

9 

Diff. 

580 

763428 

763503 

763578 

763653 

763727 

763802 

763877 

763952 

764027 

764101 

75 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4G24 

4699 

4774 

4848 

75 

2 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

3 

5669 

5743 

5818 

5892 

5P66 

6041 

6115 

6190 

6264 

6338 

74 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

5 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

.  7823 

.  74 

6 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

590 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

:  74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

8348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802 

72 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 

9596 

9669 

9741 

9813 

9885 

9957 

780029 

780101 

780173 

780245 

72 

3 

780317 

780389 

780461 

780533 

780605 

780677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

785401 

785472 

785543 

785615 

785686 

785757 

785828 

785899 

785970 

71 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

774* 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

5 

8875 

8946 

9016 

-9087 

9157 

9228 

9299 

9369 

.9440 

9510- 

71 

6 

9581 

9651 

9722 

9792 

9863 

9933 

790004 

790074  790144 

790215 

70 

7 

790285 

790356 

790426 

790496 

790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

9  1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792882  792952 

793022 

70* 

-1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

70 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

7 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890, 

69 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134, 

9203 

9272 

69 

630 

799341' 

T99409 

799478  799547 

799616 

799685 

799754 

799823 

799892 

799961 

69 

1 

800029 

800098 

800167 

800236 

800305 

800373 

800442 

800511 

800580 

800648 

69 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

69 

3 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

69 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

68 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

68 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

68 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

~m. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


11 


So. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

640 

806180,806248 
6858  6926 

806316 
-  6994 

806384J  806451 
7061  7129 

806519 
7197 

806587 
7264 

806655 
7332 

806723 
7400 

806790 
7467 

68 
68 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

3 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

67 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

67 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

810031 

810098 

810165 

67 

G 

810233 

810300 

810367 

810434 

810501 

810569 

810636 

0703 

0770 

0837 

67 

0901 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

6.50 

812913 

812980 

813047 

813114 

813181 

813247 

813314 

813381 

813448 

813514 

67 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

6445 

5511 

66 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

3175 

66 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

677f 

6838 

66 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

819610 

819676 

819741 

819807 

819873 

819939 

820004 

820070 

820136 

66 

1  820201  820267 

820333 

820399 

820464 

820530820595 

0661 

0727 

0792 

66 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

66 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

-2626 

2691 

2756 

65 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

66 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

6296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

575r 

6815 

5880 

6945 

6010 

65 

670 

826075 

826140 

826204 

826269.826334 

826399 

826464 

826528 

826593:826658 

65 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240  7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

78861  7951 

66 

3 

8015 

8080 

8144 

82Q9 

8273 

8338 

8402 

8467 

8531  8595 

64 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175  9239 

64 

5 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818  9882 

64 

6 

9947 

830011 

830075 

830139 

S3<>_M4 

sso-f;8 

830332 

830596,830460830525 

64 

7 

830589 

0653 

0717 

078JL 

0845 

09^)9 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

.  1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

832573 

832637 

832700 

832764 

832828 

832892 

832956 

833020 

833083 

64 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784 

3848 

3912, 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

63 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

872» 

8786 

63 

690 

838849 

838912 

838975 

839038 

839101 

839164 

839227 

839289 

839352 

839415 

63 

1 

9478 

9541 

9604 

96671  9729 

9792 

98551  9918!  9981,840043 

63 

2 

840106 

840169 

840232 

840294  840357 

840420  840482 

840545840608 

0671 

63 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

63 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

62 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

62 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

62 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

62 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

62 

No. 

0 

1 

2     » 

4    5 

6 

7 

8 

9 

~m 

12 


TABLE  I.      LOGARITHMS  OF  NUMBERS. 


0. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Biff. 

700 

845098 

845160 

845222 

845284 

845346 

845408 

845470 

845532 

845594 

845656 

62 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

6 

8805 

8866 

8928' 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

8 

850033 

850095 

850156 

850217 

850279 

850340 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851503 

851564 

851625 

851686 

851747 

851809 

61 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

860038 

860098 

860158 

860218 

860278 

60 

5 

860338 

860398 

860458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

730 

863323 

863382 

863442 

863501 

863561 

863620 

863680 

863739 

863799 

863858 

59 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

8 

8056 

8115 

8174 

£233 

8292 

8350 

8409 

8468 

8527 

8586 

59 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

*9114 

9173 

59 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59 

1 

9818 

9877 

9935 

9994 

870053 

870111 

870170 

870228 

870287 

870345 

59 

2 

870404 

870462 

870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

53 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

C 

2739 

2797 

2855 

2913 

.  2972 

3030 

3088 

3146 

3204 

3262 

58 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875524 

875582 

58 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

i 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

-S349 

8407 

8464 

57 

( 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

8 

9669 

9726 

9784 

9841 

9898 

9956 

880013 

880070 

880127 

880185 

57 

9 

880242 

880299 

880356 

880413 

880471 

880528 

0585 

0642 

0699 

0756 

57 

Ho. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


13 


Ho. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

760 

880814 

880871 

880928 

880985  881042 

881099  881156 

881213 

881271 

881328 

67 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

67 

3 

2525 

2381 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

57 

5 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

57 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

57 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

67 

8 

6361 

6418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

66 

770 

886491 

886547 

886604 

886660 

886716 

886773 

886829 

886885 

886942 

886998 

66 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

56 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

66 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

66 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

66 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

66 

6 

9862 

9918 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890365 

66 

7 

890421:890477890533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

66 

8 

0980  1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

66 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

66 

780 

892095 

892150 

892206 

892262 

892317 

892373 

892429 

892484 

892540 

892595 

66 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

66 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

66 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

66 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

65 

1 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

66 

G 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

66 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

66 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897627 

897682 

897737 

897792 

897847 

897902 

897957 

898012 

898067 

898122 

55 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

65 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

4 

9821 

9875 

9930 

9985 

900039 

900094 

900149 

900203 

900258 

900312 

55 

5 

900367 

900422 

900476 

900531 

0586 

0640 

0695 

0749 

0804 

0859 

65 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

903090 

903144 

903199 

903253 

903307 

903361 

903416 

903470 

903524 

903578 

54 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

64 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

64 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

64 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

64 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

64 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

64 

810 

908485 

908539 

908592 

908646 

908699 

908753 

908807 

908860 

908914 

908967 

64 

1 

9021 

9074 

9128 

9181 

9235 

9342 

93961  94491  9503 

64 

2 

9556 

9610 

9663 

9716 

9770 

$B23 

9877 

9930  :  9984,910037 

63, 

3 

910091 

910144 

910197 

910251 

910304  910358 

910411 

910464,910518 

0571 

53 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

63 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

63 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

63 

7 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

63 

8 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

63 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

63 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


Ho. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

riff. 

820 

913814 

913867 

913920 

913973 

914026 

914079 

914132 

914184 

914237 

914290  53 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

53 

3 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

53 

4 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

919549 

-  52 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

920019 

920071 

52 

2 

920123 

920176 

920228 

920280 

920332 

920384 

920436 

920489 

0541 

0593 

52 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

52 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

924331 

924383 

924434 

924486 

924538 

924589 

924641 

924693 

924744 

52 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

52 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

52 

3 

5828 

5879 

5931 

5982 

6034 

•  6085 

6137 

6188 

6240 

6291 

51 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

51 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

850 

929419 

929470 

929521 

929572 

929623 

929674 

929725 

929776 

929827 

929879 

51 

1 

9930 

9981 

930032 

930083 

930134 

930185 

930236930287 

930338 

930389 

51 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

51 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

51 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

5 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

439Y 

4448 

51 

860 

934498 

934549 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

50 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

50 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

5 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

939569 

939619 

939669 

939719 

939769 

939819 

939869 

939918 

939968 

50 

1 

940018 

940068 

940118 

940168 

940218 

940267 

940317 

940367 

940417 

940467 

50 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

50 

6 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

6 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

9 

3989 

4038 

4088 

4137 

-  4186 

4236 

4285 

4335 

4384 

4433 

49 

No! 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE  I.      LOGARITHMS  OF  NUMBERS. 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

-  8 

9  " 

Diff. 

880^944483 

944532 

944581 

944631 

944680 

944729 

944779 

944828 

944877 

944927 

49, 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49 

I 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

49 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

5 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

6 

7434 

7483 

7532 

7581 

7630 

7679 

"7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

8 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890949390 
1|  9878 
2950365 

949439949488 
9926  9975 
950414  950462 

949536949585 
950024950073 
0511   0560 

949634 
950121 
0608 

949683  949731 
9501701950219 
0657  0706 

949780 
950267 
0754 

949829 
950316 
0803 

49 
49 
49 

3!  0851 

0900  0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

4 

1338 

1386   1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

954628 

C54677 

48 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

48 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7060 

48 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

7 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990  8038 

48 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

959041 

959089 

959137 

959185 

959232 

959280 

959328 

959375I959423;959471 

48 

1 

9518 

9566 

9614 

9661 

9709 

9757 

9804  9852  9900|  9947 

48 

2 

9995 

960042 

960090 

960138 

960185 

960233 

960280:  960328  960376 

t»;i4-.'3 

48 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

08041  0851 

0899 

48 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

5   1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

6 

1895 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

9  3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920'963788 

963835 

963882 

963929 

963977 

964024 

964071 

964118 

964165 

964212 

47 

1 

4260  4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

41 

2 

4731 

4778 

4825 

4812 

4919 

4966 

5013 

5061 

5108 

6155 

47 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484  5531 

5578 

5625 

47 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954  6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517i  6564 

47 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986<  7033 

47 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454  i  7501 

47 

8 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

.  7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

968483 

968530 

968576 

968623 

968670 

968716 

968763 

968810968856968903 

47 

1 

8950 

8996 

90431  9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

2  9416 
3  9882 

9463 
9928 

9509  9556 
9975!  970021 

9602 
970068 

9649 
970114 

9695 
970161 

9742 
970207 

9789 

970254 

9835 
970300 

47 
47 

41970347 

970393 

970440 

0486 

0533 

0579 

0626 

0672  0719 

0765 

46 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

46 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

No. 

O 

1 

2 

3 

4 

5 

6    7 

8 

9 

Diff. 

16 


TABLE  I.      LOGAKITHMS  OF  NUMBERS. 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

iff. 

940 

973128 

973174 

73220 

73266 

73313 

73359 

73405 

)73451 

J73497 

973543 

46 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

6 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

g 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

977769 

77815 

77861 

77906 

77952 

77998 

978043 

978089 

978135 

46 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

5 

80003 

980049 

980094 

80140 

80185 

80231 

80276 

980322 

980367 

980412 

45 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

82271 

982316 

982362 

982407 

982452 

982497 

82543 

982588 

982633 

982678 

45 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

6 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

6741 

5786 

5830 

45 

g 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

/970 

986772 

986817 

986861 

986906 

986951 

986996 

987040 

987085 

987130 

987175 

45 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

( 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

*3 

9895 

9939 

9983 

990028 

990072 

990117 

990161 

990206 

990250 

990294 

44 

8 

990339 

990383 

990428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

991270 

991315 

991359 

991403 

991448 

991492 

991536 

991580 

991625 

44 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

• 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

» 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

j 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

( 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

j 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

6108 

5152 

•  44 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

6547 

6591 

44 

990 

995635 

995679 

995723 

995767 

995811 

995854 

995898 

995942 

995986 

996030 

44 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

i 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

! 

6948 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

] 

7386 

7430 

7474 

7517 

756 

7605 

7648 

7692 

7736 

777 

44 

j 

7823 

7867 

7910 

7954 

799 

804 

8085 

8129 

8172 

821 

44 

I 

825S 

8303 

8347 

8390 

8434 

847 

8521 

8564 

8608 

865 

44 

869E 

8739 

8782 

8826 

886 

891 

8956 

9000 

904S 

908 

44 

1 

9131 

9174 

9218 

926 

930 

934 

9392 

9435 

947£ 

952 

44 

956£ 

9609 

9652 

9696 

973 

978. 

9826 

987C 

991J 

995 

43 

No 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

riff. 

•  TABLE  II. 


NATURAL  SINES  AND  COSINES. 


18        TABLE  II.      NATURAL  SINES  AND   COSINES. 


ine.l  Cos. 


0000 
0029 
0058 
0087 
0116 
0145 
0175 
0204 
0233 
0262 
K)291 
X)320 
W349 
0378 
)0407 
K)436 
X)i65 
)0495 
)0524 
)0553 
W582 
K)611 
)0640 
)0669 
W698 
X)727 
)0756 
30785 
00814 
00844 


One. 
One. 
One. 
One. 
One. 
One". 
One. 
One. 
One. 
One. 
One. 


99999 


.99998 


.99997 


.99996 


.99996 
009021.99996 
00931  .99996 
00960  .99995 


01018  .99995 
01047  .99995 
01076  .9999; 
01105  .9999 
01134  .9999 
01164  .99991 
01193 


01222 
01251 
01280 
01309 
01338 
01367 
01396 
01425 
01454 
01483 
01513 
01542 
,01571 
,01600 
.01629 
.01658 
.01687 
.01716 
.01745 


9999, 
9999 
9999 


9999 


.9998 
.9998 
.999? 


Cos.  i  Sin 
89~ 


ne.  Cos. 


1745  .99985 
17741.99984 
18031.99984 
1832  .99983 
1862  .99983 
1891  .99982 
1920'.  99982 
1349. 00981 


Sine.  Cos. 


1978 
2007 
2036 

2065 


.09980 


.99979 


6\JVO    .  J7i7£Mi7 

209 11. 99978 
21231. 99977 
2152'.99977 
21811.99976 
2211  .99976 
2240  .99975 
2269,.  99974 
2298  .99974 
2327!.  99973 
2356  .99972 
02385 ! .99972 
32414  .99971 
32443  .99970 
02472 !.  99969 
025011.99969 
32530  .99968 
325601. 99967 
32589;.  99966 
02618  .99966 
92647  .„„„„„ 
02676  .99964 
02705  .99963 
02734  .99963 
02763  .99962 
02792  .9996: 
02821  . 99961 
02850  .9995! 
02879  .99951 
02908  .9995! 
02938  .9995 
02967  .9995 
02996  .9995 


,03490 

.03519 

.03548 

.03577 

03606 

03635 

03664 

03693 

3723 

3752 

3781 

3810 

3839 

3868 

3897 

392G 

3955 


03025 
03054 

03083 
03112 
03141 
03170 
03199 
03228 
03257 


.9995 
.9995 

.9995 
.9995 
.9995 
.9995 
.9994 


.9994 


03286  .9994 
033161.9994 
,03345!.  9994 
.03374^.9994 
.03403  .9994 
.034321.9994 
.  03461  j.  999^ 


Cos.  I  Sin 
88" 


Sine.'  Cos. 


.05234  .99863 


.99936 
.99935 


05263 
.05292 
.05321 
.05350 
.05379 


.99933 


,99931 


34013 
34042 
4071 
34100 
04129 
34159 
34188 
34217 
04246 
3427E 
04304 
04333 
04362 
04391 
0442( 
0444C 
0447* 
04507 
04536 
04565 
04594, 
04623 
04653 
04682 
04711 
04740 
,04769 
,04798 

,04827 

,04856 

.04885 

.04914 

.0494J 

.0497', 

.05001 

.0503C 

.05058 

.05088 

.0511 

.05146 

.05171 

.05205 

.05234 


99929 
99927 
99926 
99925 
99924 
99923 
99922 
99921 
,9991£ 
,99918 


.99861 


.99858 
.99857 
.99855 


4 


Sine,  i  Cos. 


99756 
99754 
99752 


.05408  .99854' 
.05437:.  99852 
.05466  .99851 
.05495  .99849 
.  05524  j.  99847 
.05553  .99846 
.  05582 1. 99844 
,05611  .99842 


.05640 


.99841 


,99916 
,99915 
.99913 
.99912 
.999111 
.99910 
.99909 
.99907 
.99906 


'.05727|.99836 
.05756;.  99834 
05785 !.  99833 
.05814  .99831 
.05844  .99829 
.05873:.  99827 
.05902  .99826 
.05931  .99824 
.05960  .99822 
05989  .99821 
06018  .9981S 


99904 
99902 
99901 
99900 
99898 
99891 
99896 
99894 


,9987, 
.9987 
.9987 
.9987 
.9986' 
.9986 
.9986 
.99864 


Cos.  Sine 
87' 


06047 
06076 
06105 
06134 


.99817 
.99811 
.9981; 
.9981! 


.06970 
.07005 
.07034 
.07063 
.07092 
.07121 
07150 
07179 
07208 
07237 
07266 
37295 
07324 
07353 
07382 
07411 
0744C 
07469 
07498 
07527 
07556 
0758 
0761 
0764. 
07672 
07701 
07730 
07759 


06163 

06192 

06221 

06250  .„ 

06279  .9980 

06308 

0633 


06366 


06424 
06453 
.06482 
,06511 
.06540 
06569 


.06627 
.06656 


_____ 
.9979 
.9979 
9979 
.9979 
.9979( 
.9978 
.9978 
.9978 
.9978 
.99781 
.9977 


06714 
.06743 
.06773 
.06802 
.06831 
.06860 
.06889 
.06918 
,06947 
.06976 


.99746 
.99744 


.9974C 

.99738 

.99736 

.99734 

.9973 

.99729 

.9972 


9972! 


.99719 
.99716 
.99714 
.99712 
.99710 
.99708 
.99705 
99703 
99701 


M. 

60 
59 
58 
57 
56 
55 
54 
53 
52 
61 
GO 
49 
48 
47 
46 
45 
44 
43 
12 
41 
40 


07817 
07846 
,07875 
07004 
07<J3iJ 
.07962 
.07991 
.08020 
.08049 
.08078 
.08107 
.08136 
.0816E 
.08194 
.08223 
.08252 
.08281 
.08310 
.0833S 
.0836 
.08397 
9977  .08426 
.99774  .08455 
99772;.  08484 
;99770:.085i: 
99768  .0854! 
;99766j.0857 
99764  .0860( 
1  99762  1.08621 
.99760  .0865! 
.997581.0868 
.997561.0871 


99696 


9968< 


,99671 
,9967: 

.9966! 


,9965< 
.99657 
.9965 
.9965 
.9964 
.9964 
.9964 


9963 


.9963 


,9962 
,9962 
.9961 


Cos.    Sine.   Cos.    Sine 
86° 


37 
36 
35 

34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 

r 


14 


M 


TABLE   II.      NATURAL  SINES  AND  COSINES. 


19 


Sine.  Cos.  Sine.  Cos. 


.10453 
.10482 
.10511 
.10540 


99452 
99449 
99446 
99443 
99440 
99437 
99434 
99431 
99428 
99424 
99421 
99418 
99415 


12181 
12216' 
12245 
12274 
12302 
12331 
12360 
12389 
12418 
12447 
12476 
12504i 
12533 
12562! 
125911 


.10597 
.10626 
.10655 

.10684 


.995941.10742 
.99591  .10771 


.16103, 
.WISH 
.16160 
.161891 
.16218 
.16246 
.16275 
.16304 
.16333 
.16361 
.16390 
.16419 
.16447 
. 16476 ! 
.16505 


.09179 
.09208 
.09237 
.09266 


99575 
99572 
99570 
995(17 


10945 
10973 
11002 
11031 

11060 


.09324 
.09353 
.09382 
.09411 
.09440 


11118 
11147 
11176 
11205 
11234! 
11263' 
11291! 
11320 
11349 
11378 
11407 
11436' 
11465 
11494 
11523 
11552 
11580; 
11609! 
11638J 
11667! 
11696: 
11725' 
117541 

11783 
11812 
11840 
11869 


99566 

9955:; 
99551 
99548 
99545 
99542 
;»:•.->  in 


99377 
99374 
99370 
99367 


'.09498 
.09527 
.09556 
.09585 
.09614 
.09642 
.09671 
.097001 
.09729 
.09758 
.09787 
.09816 
.09845 
.09874 
.09903 
.09932 
.09961 


99156  .14695 
99152  .14723 


99344 
99341 
99337 
99334 
99331 
99327 
99324 
99320 
99317 
99314 
99310 


99523 

99520 
99517 
99514 
995111 
99508 
99506 
99503 


.16706 
.16734 
.16763 
.16792 
.16820 
.16849 
.16878 
.16906 
.16935 


9SS.-4 
98849 
96846 
96841 
98836 


.10019 
.10048 
.10077 
.10106 
.10135! 
.10164 
.10192! 
.10221 
.10250 
.10279! 
.10308! 
.10337 


13514 
13543 
13572 
13600 


.15241 
.15270 
.15299 
.15827 
.15356 
.15385 
•15414 
.15442 
.15471 
.15500 
.15529 
.15557 
.15586 
.15615 
.15643 


.17021 
.17050 
.17078 
.17107 
.17136 
.17164 
.17193 
.17222 
.17250 
.17279 
.17308 
,17336 
,17365 


.99482 

,9947 

.9947 

.  99473 ; 

.99470 

.99467 

.99464 

.99461J 

i 99458 

.99455 

.99452 


11927 
11956 i 
,11985! 
,12014! 
,12043 
.12071! 
.121001 
.12129 
.12158 
.12187 


98516 
98511 
98506 
98501 
98496 


99055 

99051 
99047 
90043 


98796 
98791! 
98787 
98782 
98778 
98773 


13744 
13773 
13802 
13831 


20 


TABLE  ii.     NATURAL  SINES  AND  COSINES. 


1O° 

11° 

12° 

13° 

14° 

M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

17365 

.98481 

.19081 

.98163 

.20791 

.97815 

.22495 

97437 

24192 

97030 

60 

1 

17393 

.98476 

.1U109 

.98157 

.20820 

.97809 

.22523 

97430 

24220 

97023 

59 

2 

17422 

.98471 

.19138 

98152 

.20848 

.97803 

.22552 

97424 

24249 

97015 

58 

3 

17451 

.98466 

.19107 

98146 

.20877 

.97797 

.22580 

97417 

24277 

97008 

57 

4 

17479 

.984G1 

19195 

98140 

.20905 

.97791 

.22608 

97411 

24305 

.97001 

56 

5 

17508 

.98455 

19224 

98135 

.20933 

.97784 

.22637 

97404 

24333 

.96994 

55 

6 

17537 

.98450 

.19252 

98129 

.20962 

.97778 

.22665 

97398 

24362 

.96987 

54 

7 

17565 

.98445 

.19281 

93124 

.20990 

.97772 

.22693 

97391 

24390 

.96980 

53 

8 

17594 

.98440 

.19309 

93118 

.21019 

.97766 

.22722 

97384 

24418 

.96973 

52 

9 

17623 

.98435 

19338 

98112 

.21047 

.97760 

.22750 

97378 

24446 

.96966 

51 

10 

17651 

.98430 

.19366 

98107 

.21076 

.97754 

.22778 

97371 

24474 

.96959 

50 

11 

17680 

.98425 

19395 

98101 

.21104 

.97748 

.22807 

97365 

24503 

.96952 

49 

12 

17708 

.98420 

.19423 

98096 

.21132 

.97742 

.22835 

97358 

24581 

.96945 

48 

13 

17737 

.98414 

.19452 

98090 

.211C1 

.9773.5 

.22863 

97351 

.24559 

.96937 

47 

14 

17766 

.98409 

.19481 

98084 

.21139 

.97729 

.22892 

97345 

.24587 

.96930 

46 

15 

17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

97338 

.24615 

.96923 

45 

16 

17823 

.98399 

.19538 

98073 

.21246 

.97717 

.22948 

97331 

.24644 

.96916 

44 

17 

17852 

.98394 

.19566 

98067 

.21275 

.97711 

.22977 

97325 

.24672 

.96909 

43 

18 

17880 

.98389 

.19595 

98061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

17909 

.98383 

.19623 

98056 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20 

17937 

.98378 

.19652 

98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

17966 

.98373 

.19680 

98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

17995 

.98368 

.19709 

98039 

.21417 

.97680 

.23118 

.97291 

.24813 

.96873 

38 

23 

18023 

.98362 

.19737 

98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

18052 

.98357 

.19766 

98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

18081 

.98352 

19794 

98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

18109 

.98347 

.19823 

98016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

18138 

.98341 

19851 

98010 

.21559 

.97648 

.23260 

.97257 

.24953 

.96837 

33 

28 

18166 

.98336 

19880 

98004 

.21587 

.97642 

.23288 

.97251 

.24982 

.96829 

32 

29 

18195 

.98331 

19908 

97998 

.21616 

.97636 

.23316 

.97244 

.25010 

.96822 

31 

30 

18224 

.98325 

.19937 

97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

18252 

.98320 

.19965 

97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

18281 

.98315 

.19994 

97981 

.21701 

.97617 

.23401 

.97223 

.25094 

.96800 

28 

33 

18309 

.98310 

.20022 

97975 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

18338 

.98304 

.20051 

97969 

.21758 

.97604 

.23458 

.97210 

.25151 

.96786 

26 

35 

18367 

.98299 

.20079 

97963 

.21786 

.97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

18395 

.98294 

.20108 

97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

97952 

.21843 

.97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

.18452 

.98283 

.20165 

97946 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

97940 

.21899 

.97573 

.23599 

.97176 

.25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

.97928 

.21956 

.97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922 

.21985 

.97553 

.23684 

.97155 

.25376 

.9672" 

18 

43 

.18595 

.98256 

.20307 

.97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

.97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16, 

45 

.18652 

.98245 

.20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

.18681 

.98240 

.20393 

.97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23825 

.97120 

.25516 

.96690 

13 

48 

.18738 

.98229 

.20450 

.97887 

.22155 

.97515 

.23853 

.97113 

.25545 

.96682 

12 

49 

.18767- 

.98223 

.20478 

.97881 

.22183 

.97508 

.23882 

.97106 

.25573 

.96675 

11 

50 

.18795 

.98218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.22268 

.97489 

.23966 

.97086 

.25657 

.96653 

8 

53 

.18881 

.98201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18938 

.98190 

.20649 

.97845 

.22353 

.974701.24051 

.97065 

.25741 

.96630 

5 

56 

.18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769 

.96623 

4 

57 

.18995 

.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798 

.96615 

3 

58 

.19024 

.98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826 

.96608 

2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

60 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882 

.96593 

0 

M. 

COS. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

COS. 

Sine; 

Cos. 

Sine. 

M. 

79" 

78° 

77° 

76° 

75° 

TABLE   II.      NATURAL   SINES   AND   COSINES. 


21 


13" 

16° 

17° 

18° 

19* 

M. 

Sine. 

Cos. 

Sine.:  Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine.  | 

Cos. 

M. 

0  .25882 

.96593 

.275641.  96126!.  29237 

.95630 

.30902 

.95106 

.32557 

.94552 

^50 

1  .25910 

.96585 

.27592  .961  18  1.29265 

.95622 

.30929 

.95097 

.32584 

.94542 

59 

2  .25938 

.96578 

.27620  .961  10  .29293 

.95613 

.30957 

.95088 

.32612 

.94533 

58 

3  .25966 

.96570 

.27648  1.96102!.  29321 

.95605 

.30985 

.95079 

.32639 

.9452b  57 

4  .25994 

.96562 

.27676  .96094  .29348 

.95596 

.31012 

.95070 

.32667! 

.945141  56 

5  i.  26022 

.96555 

.27704  .96086  .29376 

.95588 

.31040 

.95061 

.32694! 

.94504;  55 

6  .26050 

.96547 

.27731  .96078  .29404 

.95579 

.31068 

.95052 

.327221.  944951  54 

7  .2607$ 

.96540 

.27759 

.96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485  53 

8  2610" 

.96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476  52 

9  .26135 

.96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466  51 

10 

.  26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

'.  95015 

.32832 

.94457]  50 

11  i  26191 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12  '.26219 

.96502 

.27899 

.96029  .29571 

.95528 

.31233 

.94997 

.32887 

.94438 

48 

13  1.26247 

.96494 

.27927 

.96021  .29599 

.95519 

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j.  34202 

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0 

M. 

Cos. 

Sine. 

COS. 

Sine. 

Cos.  |  Sine. 

Cos. 

1  Sine. 

Cos. 

Sine. 

M. 

74' 

73'       72° 

71' 

70' 

TABLE  II.      NATURAL  SINES  AND  COSINES. 


2»°       21° 

22° 

22° 

24° 

M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine.  Cos. 

Sine.]  Cos. 

Sine. 

Cos. 

VI. 

0 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92050  . 

40874 

91355 

60 

1 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92039 

40700 

91343 

59 

2 

34257 

93949 

35891 

93337 

37515 

92697 

39127 

92028 

40727 

91331 

58 

3 

34284 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91319 

57 

4 

34311 

93929 

35945 

93316 

37569 

92675 

39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91994 

40806 

91295 

55 

6 

34366 

93909 

36000 

93295 

37622 

92653 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

53 

8 

34421 

93889 

36054 

93274 

37676 

92631 

39287 

91959 

40886 

91260 

52 

9 

34448 

93879 

36081 

93264 

37703 

92620 

39314 

91948 

40913 

91248 

51 

10 

34475 

93869 

36108 

93253 

.37730 

92609 

39341 

91936 

40939 

91236 

-50 

11 

34503 

93859 

36135 

93243 

.37757 

92598 

39367 

91925  .40966 

91224 

49 

12 

34530 

93849 

.36162 

93232 

.37784 

92587 

.39394 

91914  .40992 

91212 

48 

13 

34557 

93839  36190 

93222 

.37811 

92576 

.39421 

91902  .41019 

91200 

47" 

14 

34584 

93829 

36217 

93211 

.37838 

92565 

.39448 

91891 

.41045 

91188 

46 

15 

34612 

93819 

36244 

93201 

.37805 

92554 

.39474 

91879 

.41072 

91176 

45 

16 

34639 

93809 

36271 

93190 

.37892 

92543 

.39501 

91868 

.41098 

91164 

44 

17 

34666 

93799 

36298 

93180 

.37919 

92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

34694 

93789 

36325 

93169 

.37946 

92521 

.39555 

,91845 

.41151 

.91140 

42 

19 

34721 

93779 

36352 

93159 

.37973 

92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

34748 

93769 

36379 

93148 

.37999 

92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

34775 

.93759 

36406 

93137 

.38026 

92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

34803 

.93748 

36434 

93127 

.38053 

92477 

.39661 

.91799 

.41257 

.91092  38 

23 

34830 

.93738 

36461 

93116 

.38080 

92466 

.39688 

.91787 

.41284 

.91080 

37 

24 

34857 

.93728 

36488 

93108 

.38107 

92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

34884 

.93718 

36515 

93095 

.38134 

92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

34912 

.93708 

.36542 

93084 

.38161 

92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

34939 

.93698 

.36569 

93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

34966 

.93688 

.36596 

93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

34993 

.93677 

.36623 

93052 

.38241 

.92399 

.39848 

.917181.41443 

.91008 

31 

30 

35021 

.93667 

.36650 

93042 

.38268 

.92388 

.39875 

.91706  .41469 

.90996 

30 

31 

35048 

.93657 

.36677 

93031  .38295 

.9237 

.39902 

.91694  .41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

93020 

.38322 

.9236 

.39928 

.91683 

.4152 

.90972 

28 

33 

.35102 

.93637 

.36731 

93010 

.38349 

.9235 

.39955 

.91671 

.4154 

.90960 

27 

34 

.35130 

.93626 

.36758 

92999 

.38376 

.9234 

.39982 

.91660 

.4157 

.90948 

26 

35 

.35157 

.93616 

.36785 

92988 

.38403 

.9233 

.4000 

.91648 

.4160 

.90936 

25 

36 

.35184 

.93606 

.36812 

92978 

.38430 

.9232 

.40035 

.91636 

.4162 

.90924 

24 

37 

.35211 

.93596 

.36839 

92967 

.38456 

.9231 

.4006 

.91625 

.4165 

.909111  23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.4008 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.3689 

.92945 

.38510 

.92287 

.4011 

.91601 

.41707 

.90887 

21 

40 

.3529 

.93565 

.3692 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

4t 

.3532 

.93555 

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.92924 

.3856 

.92265 

.40168 

.91578 

.41760  .90863 

19 

42 

.3534 

.93544 

.3697o 

.92913 

.3859 

.92254 

.40195 

.91566 

.41787  .90851 

18 

43 
44 

.3537 
.3540 

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.37002 
.37029 

.92902 
.92892 

.3861 
.38644 

.92243 
.92231 

.40221 

.40248 

.91555 
.91543 

.41813  .90839  17 
.41840.90826  16 

45 

.3542 

.93514 

.37056 

.92881 

.3867 

.9222C 

.40275 

.91531 

.41866 

1.90814 

15 

46 

.35456 

.93503 

.3708 

.92870 

.3869 

.9220£ 

.40301 

.91519 

.41892 

.90802 

14 

47 

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.93493 

.3711 

.92859 

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.92198 

.40328 

.91508 

.4191S 

.90790;  13 

48 

.35511 

.93483 

.3713 

.92849 

.3875 

.9218( 

.40355 

.91496 

.41945 

.90778 

1  12 

49 

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.41972 

.9076C 

11 

50 

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.93462 

.3719 

.92827 

.3880 

.92164 

.40408 

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.90752 

10 

51 

.35592 

.93452 

.3721 

.9281G 

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I 

53 

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.35674 

.9342C 

.3729 

.92784 

.3891 

.9211< 

.40514 

.91425 

.42104 

.90704 

6 

55 

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.3732 

.92772 

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'  .40541 

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5 

56 

.3572* 

.9340C 

.3735 

.92765 

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>  .40567 

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•  4 

57 

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.9338< 

.3738 

.92751 

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>  .40594 

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c 

68 

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>  .9337< 

>  .3740 

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i  .40621 

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>  2 

59 

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*  .37434 

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>  .40647 

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.9064: 

S  1 

60 

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r  .9335! 

?  .3746 

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(  .3907 

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.90631 

0 

M 

Cos. 

Sine 

COS 

Sine 

COS 

Sine 

COS. 

Sine 

Cos. 

Sine 

M, 

68s 

68* 

67° 

66° 

65  • 

TABL.&  iL      NATURAL  SINES  AND  COSINES. 


23 


25° 

26° 

27° 

28° 

2»« 

M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine.  Cos. 

Sine. 

Cos. 

Sine._  Cos. 

M. 

0 

.42262 

.90631  .43837 

.898791.45399  .89101' 

.46947  .88295 

.48481  '.87462 

60 

1 

.42288 

.90618j.  43863 

.898671.  45425;.  89087 

.46973  .88281 

.48506i.87448 

59 

2 

.42315 

.906061.43889 

.89854  1.45451!.  89074 

.46999 

.88867 

.48532  .87434 

58 

3 

.42341 

.905941.43916 

.89841  .45477  '.89061 

.47024  .88254 

.48557  .87420 

57 

4 

.42367 

.90582  .43942 

.89828'.  45503  .89048 

.470.oO:.88240 

.485831.87406 

56 

5 

.42394 

.905691.43968 

.89816  .45529  .89035 

.470761.88226 

.486081.87391 

55 

6 

.42420 

.90557  .43994  .89803  .45554  .89021 

.47101  .88213 

.48634  .87377 

54 

7 

.4244G 

.905451.44020  .89790:  .45580!.  89008 

.47127 

.88199 

.48800 

.87363 

53 

8 

.42473 

.90532  .44046 

.89777  .45606  .88995 

.47153 

.88185 

.486841.87349 

52 

9 

.42499 

.90520 

.44072 

.897645.45632 

.88981 

.47178 

.88172 

.48710  .87335 

51 

10 

.42525 

,90507 

.44098 

.  89752  >.  45658 

.88968 

.47204 

.88158 

.48735  .87321 

50 

1] 

.42^2 

.90495  .44124 

.  89739  '.  45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.904831.  44151 

.897261.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470  :.44177 

.89713 

.45736 

.88928 

.47281 

.88117  .48811 

.87278 

47 

14  .42631 

.90458  .44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446  .44229 

.89687 

;  45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48S88 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610  .45942 

.S>x±J 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968  .88808 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.459941.88795,47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020  .88782 

.47562 

.879651.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951  1.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937  .49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923  .49166 

.87079 

33 

23 

.42999 

.90284 

.44568 

.89519 

.46123  .88728 

.47665 

.87909  .49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506  .46149  .88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493  .46175  .88701 

.47716 

.87882 

.49242 

.8703C 

30 

31 

.43077 

.90246 

.44646 

.89480  .46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467  '.46226 

.88674 

.47767 

.87854 

.49293 

.87007  28 

33 

.43130 

.90221 

.44098 

.89454  .46252 

.88661 

.47793 

.87840 

.49318 

.86993  27 

34 

.43156 

.90208 

.44724 

.89441  .46278 

!  88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428^.46304  .88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43309 

.90183 

.44776 

.89415  .46330  .88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.894021.46355 

.88607J.  47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770  .49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.877561.49470 

.8690C 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743  .49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.  89350  !.  46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42  1.43366 

.90108 

.44932 

.89337!.  46484 

.88539 

.48022 

.87715 

.4954G 

.8686C 

18 

43  .43392 

.90095 

.44958 

.89324  .46510 

.88526 

.48048 

.877011.49571 

.8684C 

17 

44  .43418 

.90082 

.44984 

.893111.46536 

.88512 

.48073 

.87687 

.4959C 

.86834 

16 

45  .43445 

.90070 

.45010 

.89298  .46561 

.88499 

.48099  .87673 

.48622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285  .46587 

.88485 

.48124  .87659 

.49647 

.86805 

14 

47 

.43497 

.900451.45062  .  89272  ;.  46613  1.88472 

.48150  .87615  .49672 

.867C1 

13 

48 

.43523 

.90032 

.  45088  .  89259  .  46639i  .  88458 

.  48175;.  876C1  1.49697 

.86777 

12 

49 

.43549 

.90019 

.45114  .89245  .46664!.  88445 

.4820H.87G17  .4972C 

.867C2 

11 

50 

.43575 

.90007 

.45140  .89232  .46690!  .88431 

.48226  .8760,?  .49748 

.86748 

10 

51 

.43602 

.89994 

.45166  .89219  .46716  .88417 

.48252  .87589  .49773 

.86733 

9 

52 

.43628 

.89981 

.45192'.  89206.  46742!.  88404 

.482771.87575  .49798 

.86719 

8 

53 

.43654 

.89968  .45218  .89193'.  467671.  88390 

.483031.87561 

.49824 

.86704 

7 

54 

.43680 

.  89956  .  45243  .  891  80  .  46793  .  88377 

.483281.87546  .49849 

.86690 

6 

|55 

.43706 

.89943  .45269,  .89167  .46819:  .88363 

.48."54  .87532  .49874 

.86675 

5 

56 

.43733 

.89930;.45295  .89153  .46844  .88349 

.48379:.  87518  .49899 

.86661 

4 

57 

.43759 

1.89918!.  45321'.  89140'.  46870  .88336 

.48405  .87504  .40924 

.86646 

3 

58 

.43785 

.899051.45347  .89127;  .46896  .88322 

.484301.87490  .4C9EO 

.86632 

2 

69 

.43811 

.89892  .45373  .89114  .46921  .88308 

.48456  .87476  .49975 

.86617 

1 

60 

.43837 

.  89879  .  45399  .  89101  ;  .  46947  .  88295 

.484811.87462  50000 

.86603 

0 

M. 

Cos. 

1  Sine. 

Cos.  1  Sine. 

Cos.  i  Sine. 

Cos.  i  Sine. 

Cos. 

Sine. 

M. 

64C 

63° 

62' 

61         60 

24:        TABLE   II.      NATURAL  SINES  AND  COUINES. 


SO0 

3f 

32° 

33° 

34- 

M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine.  Cos. 

M. 

0 

.50000 

.86603 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

..50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.54513 

.83835 

.55968 

.82871  58 

3 

.50076 

.86559 

.51579 

.85672 

.53066 

.84759 

.54537 

.83819 

.55992 

.82855  57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806 

54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697 

.54635 

.83756 

.56088 

.82790 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757 

,51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.836(30 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280  .82659 

45 

16 

.50403 

.86369 

.51902 

.85476 

.5338G 

.84557 

.548541.83613 

.563051.82643 

44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.50178 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.50377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

>86266 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528 

37 

2t 

.50603 

.86251 

.52101 

*  85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511 

36 

23 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

.506.54 

86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.565451.82478 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569  .82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

J  55242 

.833C6 

.56689 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784  .82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473  |.85127 

.53951 

.84198 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498'.  85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522  .85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547  .85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572  .85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.65557 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959  .54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

61 

.51279 

.85851 

.52770 

.  84943  !.  54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928  .54269 

.83994 

5572(5 

.83034 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913  .54293 

83978 

55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897  .54317 

83962 

55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.  84882  j.  54342 

83946 

.55799 

82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866  .54366 

83930 

.55823 

82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851  .54391 

.83915 

.55847 

82953 

.57286  .81965 

3 

58 

.51454 

.85747 

.52943 

.84836  .54415 

.83899 

55871 

82936 

57310  .81949 

2 

59 

.51479 

.85732 

.52967 

.84820  .54440 

.83883 

55895 

82920 

57334  .81932 

1 

60 

.51504 

.85717 

.52992 

.84805^54464 

.83867 

55919 

.82904 

573581 

.81915 

0 

5T 

COS. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

COS. 

Sine. 

Cos. 

Sine. 

57 

59° 

58° 

57° 

56' 

55' 

TABLE  IL      NATURAL  SINES  AND  COSINES.        25 


35« 

36° 

37* 

38* 

39* 

M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

CfB. 

M. 

0 

.57358 

.81915 

.58779 

.809021 

.60182 

79864 

.61566 

.78801 

.62932 

.77715 

60 

1 

.57381 

.81899 

.58802  .80885' 

.60203 

.79846  G1689 

.78783  .62955 

.77696 

59 

2 

.57405 

.81882 

.58826  .80867J 

.60228  .79829  .61612 

.78765  .62977 

.77678 

58 

3 

.57429 

.81865 

.58849  .80850 

.60251..  79811  ;.61635 

.78747 

.63000  .77660 

57 

4  .57453 

.81848 

.58873  .80833 

.  60274  1.  79793  .61658 

.78729 

.63022  .77641 

56 

5 

57477 

.81832 

.588961.  808  16j 

.602981.79776  .61681 

.78711 

.63045  .77623 

55 

6 

57501 

.81815 

.  58920!.  80799j 

.G032H.79758I.61704 

.78694 

.630681.77605 

54 

7 

57524 

.81798 

.  58943  |.  80782 

.  60344  !.  79741 

.61726 

.78676 

.G30901.  77586 

53 

8 

57548  .81782 

.58967  .80765' 

.60367  .79723 

.61749 

.78658 

.631131.77568 

52 

9 

57572  .81765 

.58990  .80748' 

.60390;.  79706 

.61772 

.78640 

.63135  .77550 

51 

10 

575961.81748 

.59014  .80730! 

.604141.79688 

.61795 

.78622 

.63158  .77531 

50 

11 

57619 

.81731 

.  59037  j.  80713! 

.  60437  1.  79671 

.61818 

.78604 

.63180J.77513 

49 

12 

57643 

.81714 

.  59061  !.80696: 

.60460  .79653  .61841 

.78586 

.63203  .77494 

48 

13 

57667 

.81698 

.  59084  J.80679J 

.60483!  .79635!  .61864  .78568 

.63225  .77476 

47 

14 

57691 

.81681 

.59108;.  80662; 

.60506  .79618  .618871.78550 

.63248^.77458 

46 

15 

57715 

.81664 

.59131 

.80644 

.60629  .79600 

.61909 

.78532 

.63271  .77439 

45 

16 

57738 

.81647 

.59154 

.80627 

.605531.  79583 

.61932 

.78514 

.632931.77421 

44 

17 

57762 

.81631 

.59178  .80610 

.60576'.  79565  .61955 

.73496 

.63316]  .77402 

43 

18 

57786  '81614 

.59201 

.80593! 

.60599  .79547  .61978 

.78478 

.63338  .77384 

42 

19 

57810 

.81597 

.59225 

.80576' 

.606221.79530 

.62001 

.78460 

.63361 

.77366 

41 

20 

57833 

.81580 

.59248 

.80558 

.60645J.  79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

57857 

.81563 

.59272 

.80541! 

.60668  .79494  .62046 

.78424 

.63406 

.77329 

39 

22 

57881 

.81546 

.59295  .80524; 

.60691  .79477  .62069 

.78405 

63428 

.77310 

38 

23 

57904 

.81530 

.59318  .  80507  ! 

.60714!.  79459  .62092  .783871.63451 

.77292 

37 

24 

57928 

.81513 

.59342  '.80489' 

.60738  .79441 

.62115  .78369  .63473 

.77273 

36 

25 

57952 

.81496 

.59365  .80472 

.607611.79424 

.62138  .78351:.  63496 

.77255 

35 

26 

57976 

.81479 

.593891.80455 

.60784  .79406 

.62160  .78333!  .63518 

.77236 

34 

27 

57999 

.81462 

.59412 

.80438 

.608071.79388 

.62183  .78315  .63540 

.77218 

33 

28 

58023 

.81445 

.59436 

.80420 

.60830:.  79371 

.62206 

.78297  .63563 

.77199 

32 

29 

58047 

.81428 

.59459 

.80403 

.60853  .79353 

.62229 

.78279!  .63585 

.77181 

31 

30 

58070 

.81412 

.59482 

.80386 

.60876  .79335 

.62251 

.78261  .63608 

.77162 

30 

31 

58094 

.81395 

.59506 

.80368 

.60899  .79318 

.62274 

.78243  .63630 

.77144 

29 

32 

58118 

.81378 

.59529 

.80351 

.60922  .79300 

.62297 

.782251.63653 

.77125 

28 

33 

58141 

.81361 

.59552 

.80334 

.60945  .79282 

.62320  .782061.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264  .62342 

.78188  .63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247  .62365 

.78170  .63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229  .62388 

.78152  .63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211  .62411 

.78134  .63765  .77033 

23 

38 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193  .62433 

.78116|.63787  .77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.791761.62456 

.78098  .63810  .76996 

21 

40 

.58307 

.81242 

.59716 

.S0212 

.61107 

.79158-62479 

.78079  .63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.  79140;  .  62502  1  .  78061  1  .  63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122  .62524  .78043  .63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105  .625471.78025  .63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570  .78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592  .77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.790511  .62615,  .7797o!  .63966 

.76866 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268!.  790331.  62638 

.77952  .63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291.790151.62660 

.77934  .64011  .76828 

12 

49 

•58519 

.81089 

.59926 

.80056 

.61314.789981.62683 

.77916  .64033  -76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980  .62706 

.77897!.  64056,.  76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962  .62728 

.77879 

.64078  .76772 

9 

52 

.58590 

.81038  .59995 

.80003 

.61383 

.78944'  .62751 

.77861 

.641001.76754 

8 

53 

.58614  .81021  ,60019 

.79986 

.61406 

.78926  .62774 

.77843 

.64123  .76735 

7 

54 

.58637  .81004 

.60042 

.79968 

.61429 

.78908  .62796 

.77824 

.64145'.  76717 

6 

55 

.586611.80987 

.60065 

.79951 

.61451 

.  78891  ;.  62819 

.77806 

.64167  .76698 

5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873J  .  62842 

.77788 

.64190  .76679 

4 

57 

.58708 

.80953 

.60112  .79916 

.61497 

.78855;.  62864 

.77769 

64212  .76661 

3 

58 

.58731 

.80936 

.60135  .79899 

.61520 

.78837  .62887 

.77751 

.64234  .76642 

2 

59 

.58755  .80919 

.601581.79881 

.61543 

.78819  .62909 

.77733 

.64256'.  76623 

1 

60 

.58779  .80902 

.60182  '.79864 

.61566 

.78801  .62932 

.77715 

.64279'.  76604 

0 

M 

Cos.  1  Sine. 

Cos.  '  Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos.  Isine. 

M. 

54° 

53* 

52" 

51° 

50' 

26        TABLE   II.      NATURAL  SINES  AND  COSINES. 


4<r 

41'    |    42' 

43 

44° 

M. 

Sine. 

Cos. 

Sine.  Cos.  i  Sine. 

Cos. 

Sine. 

COS. 

Sine. 

009. 

• 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
11 
15 

64279 
64301 
64323 
64346 
64368 
64390 
.64412 
.64435 
.64157 
.64479 
.64501 
.64524 
.64546 
64568! 
64590 
64612 

.76604 
.76586 
.76567 
.76548 
.76530 
.76511 
.76492 
.76473 
.76455 
.76436 
.76417 
.76398 
.76380 
.76361 
.76342 
.76323 

65606 
65628 
65650 
.65672 
.65694 
.65716 
.65738 
.65759 
.65781 
65803 
65825 
65847 
65869 
05891 
65013! 
6u935 

75471 
75452 
.75433 
.75414 
.75395, 
.75375 
.75356 
.75337 
.75318 
.75299 
.75280 
.75261 
.75241 
.75222 
.7^203 
.75184 

66913 
66935! 
66956 
66978; 
66999 
67021 
67043 
67064 
67086 
67107 
67129 
67151 
67172 
67194 
67215 
67237 

74314 
74295 
74276 
74256 
74237 
74217 
74198 
74178 
74159 
.74139 
.74120 
.74100 
.74080 
.74061 
.74041 
.74022 

68200 
68221 
68242 
68264 
68285 
68306 
.68327 
68349 
.68370 
.68391 
.68412 
.68434 
.68455 
.68476 
.68497 
.68518 

73135 
73116 

73096 
73076 
73056 
73036 
73016 
72996 
72976 
.72957 
.72937 
.72917 
.72897 
.72877 
.72857 
.72337 

69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 
69633 
69654 
696751 
69696 
69717 
69737, 
69758 
69779 

71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 
71772 
71752 
71732 
71711 
71691 
71671 
.71650 
.71630 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

16 

64635 

.76304 

65956 

75165 

67258 

74002 

.68539  .72817 

69800 

.71610 

44 

17 
18 

64657  .'76286 
64679  .762SV 

65978 
66000 

751-16 
75126 

67200 
67301 

73933 
73963 

.68561  .72797 

.68582  .72777 

69821 
69842 

.71590 
.71569 

43 

42 

19 

64701 

.76248 

66022 

75107 

67323 

73944 

.  68603  j.  72757 

69862  .71549 

41 

20 
21 

64723 
64746 

.76229 
76210 

.66044 
.66066 

75088 
75069 

.67344 
.67360 

73924 
73904 

.68624  .72737 
.686451.72717 

69883!  .  71529 
69904!  .  71508 

40 

39 

22 

64768 

76192 

.6608G 

.75050 

.67387 

73885 

.68666 

.72697 

69925  .71488 

38 

23 

64790 

76173 

.66109 

.75030 

.67409 

73865 

.68688 

.72677 

69946  .71468 

37 

24 
25 
26 

64812 
64834 
64856 

76154 
76135 
76116 

.60131 
.66153 
.66175 

.75011 
.74992 
.74973 

.67430 
.67452 
.67473 

',3846 
.73826 
.73806 

.68709 
.68730 
.68751 

.72657 
.72637 
.72617 

69966 
69987 
70008 

.714*7 

.71427 
.71407 

36 
35 

34 

27 
28 

64818 
64901 

76097 

76078 

.66197 

.66218 

.74953 
.74934 

.67495 
.67516 

.73787 
.73767 

.68772 
.68793 

.72597 
.72577 

70029  .71386 
700491.71366 

33 
32 

29 

61923 

76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557 

70070 

7134& 

31 

30 

64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

70091 

71325 

30 

31 

64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517 

70112 

7130 

29 

32 

64989 

.76003 

.66306 

.74857 

.67602 

.73688 

.68878  .72497 

70132 

7128 

28 

33 

65011 

.75984 

.66327 

.74838 

.67623 

.73669 

.68899  .72477 

.70153 

71264 

27 

34 
35 
36 
37 

65033 
.65055 
.65077 
.65100 

.75965 
.75946 

l7f)J)08 

.66349 
.66371 
.66393 
.66414 

.74818 
.74799 
.74780 
.74760 

.67645 
67666 
67688 
67709 

.73649 
.73629 
.73610 
.73590 

.68920 
.68941 
.68962 
.68983 

.72457 
.72437 
.72417 
.72397 

.70174 
.70195 
.70215 
.70236 

7124 
7122 
.7120 
.7118 

26 
25 
24 
23 

38 
39 

.65122 
.65144 
65166 

.75889 
.75870 
.75851 

.66436 
.66458 
.66480 

.74741 
.74722 
.74703 

67730 
67752 
67773 

•s 

.73531 

.69004 
.69025 
.69046 

.72377 

.72357 
.72337 

.70257 
.70277 
.70298 

.7116 
.7114 
.7112 

22 
21 
20 

41 
42 

.65183 
.65210 

.7583? 
.75813 

.66501 
.66523 

.74683 
.74664 

.67795 
.67816 

73511 
.73491 

.69067 
.69088 

.72317 
.72297 

70319 
.73339 

.71100 
.7108 

19 

18 

43 

.65232 

.7579-1 

.66545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.7105 

17 

44 

.6525^ 

.75775 

.66566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.7103 

16 

45 

.65276 

.75<56 

.66588 

.74606 

.6.880 

.73432 

.69151 

.72236 

.70401 

.7101 

15 

46 

.65298 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.7221 

.70422 

.7099 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.7097 

13 

48 

.65342 

.75700 

.66653 

.74548 

.67944 

.73373 

.69214 

.7217 

.70463 

.7095 

12 

49 

.65364 

.75680 

.66675  .74528 

.67965 

.73353 

.69235 

.7215C 

.70484 

.7093 

11 

50 

K1 

.6538C 
.6540? 

.75661 
.75642 

.66697 
.66718 

.74509 
.74489 

.67987 
.68008 

.73333 
.73314 

.69256 
.69277 

.72136 
.7211 

.70  JOE 

.7052E 

.7091 
.7089 

10 
0 

OL 

52 
53 

.6543C 
.65452 

.75622 
.75604 

.6674-9!.74470 
.66762  .74451 

.6802S 
.68051 

.73294 
.73274 

.69298 
.69318 

.7209 
.7207 

.70546 
.70567 

.7087 
.7085 

8 

54 

.65474 

.7558E 

.66783  .74431 

.68072 

.732.54 

.69340 

.7205 

.70587 

.70834 

( 

55 

.65491 

.7556G 

.6680! 

>  .74412 

.68093 

.73234 

.69361 

.7203, 

.7060? 

.7081 

j 

56 

.6551? 

.75547 

.6682' 

.74392 

.69382 

.7201 

.7062? 

.7079 

< 

57 

.6554C 

.7552? 

.66848  .74373 

'.68136 

.7319E 

,  .6940C 

.7199 

.7064* 

.7077 

'< 

58 

.65562 

.7550£ 

.66870  .74353 

.6815r 

.7317f 

>  .69424 

.7197 

.7067( 

.7075 

! 

59 

,65584 

.7549( 

.6689 

L  .74334 

.68171 

.7315E 

>  .6944E 

.7195^ 

.7069( 

.7073 

60 

.6560* 

.7547] 

.6691, 

J  .74314 

.68201 

.7313, 

i  .6946< 

.7193^ 

.70711 

.7071 

M7 

Cos. 

Sine 

Cos. 

Sine 

Cos. 

Sine 

.  Cos. 

Sine 

Cos. 

Sin 

M. 

49* 

48° 

47° 

4G° 

45° 

TABLE  III. 


NATURAL  TANGENTS 


AND 


COTANGENTS. 


28 


TABLE  III.   NATUBAL  TANGENTS,  ETC. 


O° 

1° 

2° 

3' 

M' 

Tang. 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang 

M- 

0 

.00000 

Infinite 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

69 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.3711 

68 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

67 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

66 

6 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

65 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

64 

7 

.00204 

491.106 

.01949 

61.3032!     .03696 

27.0566 

.05445 

18.3655 

63 

8 

.00233 

429.718 

.01978 

60.5485      .03725 

26.8450 

.05474 

18.2677 

62 

9 

.00262 

381.971 

.02007 

49.8157       03754 

26.6367 

.05503 

18.1708 

61 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

,60 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47  7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02163 

46  4-189 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45  8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2861 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44  0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180  932 

.02298 

43  5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42  96411     .04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357!  42.4335 

.04104 

24.3675 

.05854 

17.10837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162!  24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191    23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

M.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38  1885 

.04366 

.9038 

.06116 

16.3499 

30 

31 

.00902 

110  892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16-.  1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02861 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.I5056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

,06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31  S205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455   68.7501 

.03201 

31  2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317   30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346   29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

66 

.01629 

61.3829 

.03376 

29.6245 

!.  05124 

19.5156 

.06876 

14.5438 

4 

67 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

69 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

M. 

Cotang.  I  Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

89° 

88* 

87° 

86° 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


4- 

5°           l          6- 

7° 

M 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

M. 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

12367 

8.08600  57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674,56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.0475*  55 

6 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.0284*154 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948|53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.9905852 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176!51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302J50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438  49 

12 

.07344 

13.6174 

.09101 

10  9882 

.10863 

9.20516 

.12633 

7.9158248 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895  46 

15 

.07431 

13.4566 

.09189 

10.8829 

..10952 

9.13093 

.12722 

7.8606445 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

2<> 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

2.-, 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 
28 

.07782 
.07812 

12.8496 
12.8014 

.09541 
.09570 

10.4813 
10.4491 

.11305 
.11335 

8  84551 
8.82252 

.13076 
.13106 

7.64732J33 
7.63005^32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287  31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.5957530 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.5787229 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.5617628 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.5448727 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

13284 

7.5280626 

3.-, 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132  25 

30 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.4946524 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806  23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154  22 

39 

.08134 

12.2946 

.09898 

10.1080 

.11659 

8.57718 

.13432 

7.4450921 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.4287120 

41 

.08192    12.2067 

.09952 

10.0483 

.11718 

8.53402 

.13491 

7.41240  19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616  18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999  17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389  16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786  15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190  14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600  13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018  12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442  It 

50 

.08456 

11.8262 

.10216 

9  78817 

.11983 

8.34496 

.13758 

7  26873  10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310J  9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754J  8 

53 

.08544 

11.7045 

.10305 

9  70441 

.12072 

8.28376 

.13846 

7.22204J  7 

.54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661!  6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125   5 

56 

,08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422   9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

.58 

.08690 

11.5072 

.10452   9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

M. 

Cotang. 

Tang. 

Cotang, 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

85' 

84° 

83' 

82° 

30 


TABLE  III.   NATURAL  TANGENTS,  ETC. 


8° 

9° 

1O° 

11* 

M- 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang.  1  M. 

.14054 

7  11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

6.14455  60 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658  59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862  68 

8 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

67 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

6.11279 

56 

5 

.14202 

7.04105 

116888 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

B5 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

6.61397 

.19619 

5.09704 

54 

7 

.14262 

7.01174 

.16047 

6.23160 

.17843 

6.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

6.59511 

.19680 

B.  08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

6.58573 

.19710 

B.  07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

B.  06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

6.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

25 

.14767    6.77199 
.14796   6.75838 

.16555 
.16585 

6.04051 
6.02962 

.18353 
.18384 

5.44857 
5.43966 

.20164 
.20194 

4.95945 
4.95201 

36 
35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

6.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

36 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

26 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15^13 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4  .  83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4  .  82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4  81471 

16 

45 

.15391 

6.49710 

.17183 

5  81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.75937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.752191    7 

54 

.15660 

6.38587 

.174531  6.72974 

.19257 

5.19293 

.21073 

4.74534     6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851     5 

56 

.15719 

6  36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170    4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490     3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813    2 

59 

.15808 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137!    1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

u 

M 

Cotang.i  Tang. 

Cotang. 

Tang. 

Cotang.i  Tang. 

Cotang 

Tang. 

M. 

81" 

80° 

79' 

78° 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


31 


12° 

1* 

14 

15° 

M. 

Tang. 

Cotang. 

Tang.  |  Cotang 

Tang. 

Cotang 

Tang. 

rotang. 

M. 

0 

.21256 

4.70463 

.230871  4.33148 

.24933)  4.01078 

.26795 

3.73205 

60 

] 

.21286 

4.69791 

.23117   4.32573 

.24964    4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.0008G 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.9909? 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

26951 

3.71046 

55 

C 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529   4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.215601  4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485  49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.951% 

.27169 

3.68061148 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638]  47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.592*3 

.23608 

4.23580 

.25459 

3  92793 

.27326 

3.65957  43 

18 

.21804    4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538!  42 

1'J 

.21834 

4.5NH»1 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20S42 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

b.  89945 

.27513 

3.63461 

37 

24 

.21986;  4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048  36 

25 

.22017!  4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636  35 

28 

.22047;  4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224  34 

27 

.22078    4.52941 

.23916 

4.18137 

.25769 

3>M-»,x 

.27638 

3.61814  33 

26 

.22108:  4.52316 

.23946 

4.17600 

.25800 

a.  87601 

.27670 

3.61405  32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

S.  871  36 

.27701 

3.60996  31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3.60588'  30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181   29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775  28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370  27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966  26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562  25 

3»3 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160  24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758  23 

n 

.224141  4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357  22 

99 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536'  4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.2820! 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26351 

3.79378 

.28234 

3.54179 

14 

47 

.22689   4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719;  4.40152 

.24562 

4.0712" 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.  39,560 

.24593 

4.06616 

.26452 

3.78040 

.28328 

3.53001 

11 

i 

.22781 

4.38969 

.2462-t 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.2839 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.5182S 

8 

53|     .22872 

4.37207 

.24717 

4.(H.r)S6 

.26577 

3.76268 

.28454 

3-51441 

7 

54 

.22903 

4.36623 

.24747!  4.04081 

.26608 

3.75828 

.28486 

3.51063 

6 

55 

.22934 

4.360401     .247781  4.a3578 

-.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.a>t59 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50278 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.2858C 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49508 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.4912E 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

M 

Cotang 

Tang. 

Cotang 

Tang, 

Cotang 

Tang. 

Cotang 

Tang. 

M. 

77° 

76° 

75° 

74C 

TABLE  HI.   NATURAL  TANGENTS,  ETC. 


16° 

17C 

18"                          19° 

M 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang.  j  Tang. 

Cotang 

M. 

( 

>      .2867£ 

3.4874 

.3057 

3.2708E 

.32492 

3.0776? 

.344£ 

2.90421 

60 

.2870€ 

3.4835 

.3060 

3.26745 

.32524 

3.07464 

.3446£ 

2.  aoui 

59 

.28738 

3.  4797 

.3063 

3.26406 

.32556!  3.0716C 

.34498!  2.8987£ 

68 

.2876£ 

3.4759 

.3066 

3.26067 

.32588'  3.06857 

.3453C 

2.8960C 

57 

.2880C 

3,4721 

.30700 

3.25729 

.32621 

3.06554 

.3456£ 

2.8932 

56 

.28832 

3.4683 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.8905 

65 

.28864 

3.4645* 

.30764 

3.25055 

.32685 

3.059501     .3462* 

2.8878 

5< 

.28895 

3.4608 

.30796 

3.24719 

.32717 

3.05649L     .34661 

2.8851 

63 

.28927 

3.4570 

.30828 

3.24383 

.32749 

3.05349!      .34692 

2.8824 

52 

.28958 

3.4532 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.8797 

51 

1 

.28990 

3.4495 

.3089 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

60 

1 

.29021 

3.4457 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

1 

.29053 

3.4420 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.8716 

48 

13 

.29084 

3.4382 

.3098-- 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.4345 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.8662 

46 

lo 

.2914 

3  .43084 

.31051 

3.22053 

=32975 

3.03260 

.34922 

2.86356 

45 

1C 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.8608 

M 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.8582 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3,02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266   3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298   3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330  3.00028 

.35281 

2.83439 

34 

21 

.29526 

3.38679 

.31434 

3.18127 

.33363  -2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395   2.99447 

.35346 

2.8291' 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

S.  16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.816K 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26' 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2  .  79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

,36035 

2.77507 

[1 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

LO 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2,76750 

g 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

8.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

73* 

72° 

71' 

70' 

i 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


33 


2O° 

21' 

22° 

23° 

M. 

Tang. 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2  35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

56529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.3463655 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.3444754 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.3425853 

8 

,36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.3406952 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.3388151 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.3369350 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.3350549 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317;48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.3313047 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.3294346 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.3275645 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.3257044 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2!  44027i 

.43032 

2.3238343 

1?* 

.36991 

2.70335 

.38988 

2.56486 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.3145638 

28 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271  37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.3108636 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.3090235 

20 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.3071834 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

H 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

,39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

a^ 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.SMH 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51288 

.41831 

2.39058 

.43897 

2.27806 

18 

43|     .37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44      .37853 

2.64177 

.39862 

2.50864 

.41899 

2.38688 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2  38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2  48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2.48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24556 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

.38386 

2.60509      .40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

j 

M. 

Cotang. 

Tang.  Cotang. 

Tang. 

Cotang. 

Tang.  Cotang. 

Tang. 

M. 

69* 

68* 

67° 

66* 

TABLE  IH.   NATUBAL  TANGENTS;  ETC. 


24° 

£5U 

26° 

27° 

M 

Tang. 

Cotarig 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

M. 

.44523 

2.24604 

.46631 

2.1445 

.4877 

2.0503 

.5095 

1.9626 

60 

.44558 

2.24428 

.46666 

2.1428 

.4880 

2.0487 

.5098 

1.96120 

59 

.44593 

2.24252 

.46705 

2.1412 

.4884 

2.0472 

.5102 

1.9597 

58 

, 

.44627 

2.24077 

.46737 

2.1396J 

.4888 

2.0457 

.5106 

1.9583 

57 

- 

.44662 

2.23902 

.46772 

2.1380 

.4891 

2.0442 

.5109 

1.9569 

50 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.9555 

55 

i 

.44732 

2.23553 

.46843 

2.1347' 

.48989 

2.04125 

.51173 

1.9541 

54 

' 

.44767 

2.23378 

.4687S 

2.13316 

.49026 

2.03975 

.51209 

1.9527 

53 

1 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.9513 

52 

c 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.450417 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94022 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

•1.93195 

38. 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.927C2 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

2° 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

60 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

65 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

6 

66 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

67 

.46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

68 

.46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

M. 

Cotang. 

Tang. 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang 

Tang. 

M. 

65° 

64° 

63' 

62° 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


35 


28" 

29° 

30° 

81- 

M. 

Tang. 

Cotang. 

Tang.  |  Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.53171 

.88073 

.55431 

1.80405 

.57735 

73205 

.60086 

.66428 

60 

1 

.53208 

.87941 

.55469 

1.80281 

.57774 

73089 

.60126 

.66318 

59 

2 

.53246 

.87809 

.55507 

1.80158 

.57813 

.72973 

.60165 

.66209 

58 

3 

.53283 

.87677 

.55545 

1.80034 

.57851 

.72857 

.60205 

.66099 

57 

4 

.53320 

.87546 

.55583 

1.79911 

.57890 

.72741 

.60245 

.65990 

56 

5 

.53358 

.87415 

.55621 

1.79788 

.57929 

.72625 

.60284 

.65881 

55 

6 

.53396 

.87283 

.55659 

1.79665 

.57968 

.72509 

.60324 

.65772 

54 

7 

.53432 

.87152 

.55697 

1.79542 

.58007 

.72393 

.60364 

.65663 

53 

8 

.53470 

.87021 

.55736 

1.79419 

.58046 

.72278 

.60403 

.65554 

52 

9 

.53507 

.86891 

.55774 

1.79296 

.58085 

.72163 

.60443 

.65445 

51 

.53545 

.86760 

.55812 

1.79174 

.58124 

.72047 

.60483 

.65337 

50 

.53582 

.86630 

.55850 

1.79051 

.58162 

.71932 

.60522 

.65228 

49 

.53620 

.86499 

.55888 

1.78929 

.58201 

.71817 

.60562 

.65120 

48 

.53657 

.86369 

.55926 

1.78807 

.58240 

.71702 

.60602 

.65011 

47 

.53694 

.86239 

.55964 

1.78685 

.58279 

.71588 

.60642 

.64903 

46 

.53732 

.86109 

.56003 

1  78563 

.58318 

.71473 

.60681 

.64795 

45 

.53769 

.85979 

.56041 

1.78441 

.58357 

.71358 

.60721 

.64687 

44 

.53807 

.85850 

.56079 

1.78319 

.58396 

.71244 

.60761 

.64579 

43 

.53844 

.85720 

.56117 

1.78198 

.58435 

.71129 

.60801 

.64471 

42 

.53882 

.85591 

.56156 

1.78077 

.58474 

.71015 

.60841 

.64363 

41 

.53920 

.85462 

.56194 

1.77955 

.58513 

.70901 

.60881 

.64256 

40 

.53957 

.85333 

.56232 

1.77834 

.58552 

.70787 

.60921 

.64148 

39 

.53995 

.85204 

.56270 

1.77713 

.58591 

.70673 

.60960 

.64041 

38 

.54032 

.85075 

.56309 

1.77592 

.58631 

.70560 

.61000 

.63934 

37 

.54070 

.84946 

.56347 

1.77471 

.58670 

.70446 

.61040 

.63826 

36 

.54107 

.84818 

.56385 

1.77351 

.58709 

.70332 

.61080 

.63719 

85 

.54145 

.84689 

.56424 

1.77230 

.58748 

.70219 

.61120 

.63612 

84 

.54183 

.84561 

.56462 

1.77110 

.58787 

.70106 

.61160 

.63505 

83 

.54220 

.84433 

.56501 

1.76990 

.58826 

.69992 

.61200 

.63398 

3^ 

.54258 

.84305 

.56539 

1.76869 

.58865 

.69879 

.61240 

.63292 

31 

.54296 

.84177 

.56577 

1.76749 

.58905 

.69766 

.61280 

.63185 

80 

.54333 

.84049 

.56616 

1.76629 

.58944 

.69653 

.61320 

.63079 

29 

.54371 

.83922 

.56654 

1.76510 

.58983 

.69541 

.61360 

.62972 

28 

.54409 

.83794 

.56693 

1.76390 

.59022 

.69428 

.61400 

.62866 

27 

3* 

.54446 

.83667 

.56731 

1.76271 

.59061 

.69316 

.61440 

.62760 

26 

35 

.54484 

.83540 

.56769 

1.76151 

.59101 

.69203 

.61480 

.62654 

25 

36 

.54522 

.83413 

.56808 

1.76032 

.59140 

.69091 

.61520 

.62548 

24 

37 

.54560 

.83286 

.56846 

1.75913 

.59179 

.68979 

.61561 

.62442 

23 

38 

.54597 

.83159 

.56885 

1.75794 

.59218 

.68866 

.61601 

.62336 

•2-2 

39 

.54635 

.83033 

.56923 

1.75675 

.59258 

.68754 

.61641 

.62230 

21 

40 

.54673 

.82906 

.56962 

1.75556 

.59297 

.68643 

.61681 

.62125 

20 

41 

.54711 

.82780 

.57000 

1.75437 

.59336 

.68531 

.61721 

.62019 

19 

42 

.54748 

.82654 

.57039 

1.75319 

.59376 

.68419 

.61761 

.61914 

18 

43 

.54786 

.82528 

.57078 

1.75200 

.59415 

.68308 

.61801 

.61808 

17 

44 

.54824 

.82402 

.57116 

1.75082 

.59454 

.68196 

.61842 

.61703 

16 

45 

.54862 

.82276 

.5.7155 

1.74964 

.59494 

.68085 

.61882 

.61598 

15 

46 

.54900 

.82150 

.57193 

1.74846 

.59533 

.67974 

.61922 

.61493 

14 

47 

.54938 

.82025 

.57232 

1.74728 

.59573 

.67863 

.61962 

.61388 

13 

48 

.54975 

.81899 

.57271 

1.74610 

.59612 

.67752 

.62003 

.61283 

12 

49 

.55013 

.81774 

.57309 

1.74492 

.59651 

.67641 

.62043 

.61179 

11 

50 

.55051 

.81649 

.57348 

1.74375 

.59691 

.67530 

.62083 

.61074 

10 

51 

.55089 

.81524 

.57386 

1.74257 

.59730 

.67419 

.62124 

.60970 

g 

52 

.55127 

.81399 

.57425 

1.74140 

.59770 

.67309 

.62164 

.60865 

8 

53 

.55165 

.81274 

.57464 

1.74022 

.59809 

.67198 

.62204 

.60761 

7 

5t 

.55203 

.81150 

.57503 

1.73905 

.59849 

.67088 

.62245 

.60657 

6 

55 

.55241 

.81025 

.57541 

1.73788 

.59888 

.66978 

.62285 

.60553 

6 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

.66867 

.62325!  1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

.66757 

.623661  1.60345 

8 

58 

.55355 

1.80653 

.57657 

1.73438 

.6000< 

.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

.66428 

.62487 

1.60033 

0 

M 

1  Cotang. 

Tang. 

Cotang.  Tang. 

Cotang. 

Tang. 

rotang. 

Tang. 

5T 

61° 

60' 

59° 

58 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


3 

3° 

2 

3° 

3 

4° 

1       * 

5° 

M 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

M- 

i 

.62487 

1.6003*. 

.6494 

1.5398 

.6745 

1.48256 

.7002 

1.4281 

60 

.62527 

1.5993C 

.6498 

1.53888 

>6749 

1,4816 

.70064 

1.4272 

59 

i 

.62568 

1.59826 

.6502 

1.5379 

.6753 

1.4807 

.7010 

1.4263 

58 

j 

.62608 

1.59723 

.6506 

1.5369 

.6757 

1.4797 

.7015 

1.4255 

57 

i 

.6264S 

1.5962C 

»  .65106 

1.5359 

.6762 

1.4788 

.7019 

1.4246 

66 

i 

.62689 

1.59517 

.6514 

1.5349 

.6766 

1.4779 

.7023 

1.4237 

65 

i 

.62730 

1.59414 

.6518 

1.5340C 

.6770 

1.4769 

.7028 

1.4228 

54 

1 

.62770 

1.5931 

.6523 

1.5330 

.6774 

1.4760 

.7032 

1.4219 

53 

.62811 

1.59208 

.6527 

1.5320 

.67790 

1.4751 

.7036 

1.4211 

62 

; 

.62852 

1.5910S 

.6531 

1.5310 

.6783 

1.4742 

.7041 

1.4202 

61 

1 

.62892 

1.59002 

.6535 

1.5301 

.6787 

1.4733( 

.7045 

1.41934 

60 

.62933 

1.58900 

.6539 

1.5291 

.6791 

1.4723 

.7049 

1.4184 

49 

.62973 

1.58797 

.6543 

1.5281 

.6796 

1.4714 

.7054 

1.4175 

48 

.63014 

1.58695 

.6548 

1.5271 

.6800 

1.4705 

.7058 

1.4167 

47 

i 

.63055 

1.58593 

.6552 

1.5262 

.6804 

1.4696 

.7062 

1.4158 

46 

15 

.63095 

1.5849C 

.6556 

1.5252 

.6808 

1.4687 

.7067 

1.4149' 

45 

16 

.63136 

1.58388 

.6560 

1.5242 

.6813 

1.4677 

.7071 

1.4140 

44 

17 

.63177 

1.58286 

.6564 

1.5233 

.6817 

1.4668 

.7076 

1.4132 

43 

18 

.63217 

1.58184 

.6568 

1.52235 

.6821 

1.4659 

.70804 

1.4123, 

42 

19 

.63258 

1.58083 

.65729 

1.5213 

.6825 

1.4650 

.7084 

1.4114 

41 

20 

.63299 

1.57981 

.6577 

1.5204 

.6830 

1.4641 

.7089 

1.4106 

40 

21 

.63340 

1.57879 

.65813 

1.5194 

.6834 

1.4632 

.7093 

1.4097 

89 

22 

.63380 

1.57778 

.65854 

1.51850 

.6838 

1.4622 

.7097 

1.4088 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.6842 

1.4613 

.7102 

1.4080C 

37 

24 

.63462 

1.57575 

.65938 

1.51658 

.6847 

1.4604 

.7106 

1.4071 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.6851 

1.4595 

.7111 

1.4062 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.6855 

1.4586 

.71154 

1.4054 

34 

27 

.63584 

1.57271 

.66063 

1.51370 

.68600 

1.4577 

.71198 

1.4045 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.4568 

.71242 

1.4036 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.4559 

.71285 

1.4028 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.4550 

.71329 

1.4019 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.4541 

.71373 

1.4010 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.4532 

.71417 

1.4002 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.4522 

.71461 

1.399E 

27 

3* 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.3985 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.3976 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.3967 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.3959 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.3950 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.3942 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1  .  49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

62 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

59 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

M^ 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Uotang. 

Tang. 

M. 

5 

7* 

5 

8° 

a 

5' 

54 

f 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


37 


36° 

37° 

38° 

39* 

M. 

Tang.  Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.72654 

1.37638 

.75355 

.32704 

.78129 

.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

.32624 

.78175 

.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

.32544 

.78222 

.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

.32464 

.78269 

.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

.32384 

.78316 

.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

.32304 

.78363 

.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

.32224 

.78410 

.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.T5675 

.32144 

.78457 

.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

.32064 

.78504 

.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

.31984 

.78551 

.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

.31904 

.78598 

.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

.31825 

.78645 

.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

.31745 

.78692 

.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

.31666 

.78739 

.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

.31586 

.78786 

.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

.31507 

.78834 

.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

.31427 

.78881 

.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

.31348 

.78928 

.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

.31269 

.78975 

.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

.31190 

.79022 

.26546 

.81898 

1.22104 

41 

20 

.7a547 

1.35968 

.76272 

.31110 

.79070 

.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

.31031 

.79117 

.26395 

.81995 

1.21959139 

22 

.73637 

1.35802 

.76364 

.30952 

.79164 

.26319 

.82044 

1.21886  38 

83 

.73681 

1.35719 

.76410 

.30873 

.79212 

.26244 

.82092 

1.21814  37 

24 

.73726 

1.35637 

.76456 

.30795 

.79259 

.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

.30716 

.79306 

.26093 

.82190 

1.21670 

35 

2.: 

.73816 

1.35472 

.76&8 

.30637 

.79354 

.26018 

.82238 

1.21598  34 

27 

.73861 

1.35389 

.76594 

.30558 

.79401 

.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

.30480 

.79449 

.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

.30401 

.79496 

.25792 

.82385 

1.21382  31 

30 

.73996 

1.35142 

.76733 

.30323 

.79544 

.25717 

.82434 

1.21310  30 

31 

.74041 

1.35060 

.76779 

.30244 

.79591 

.25642 

.82483 

1.21238  29 

82 

.74086 

1.34978 

.76825 

.30166 

.79639 

.25567 

.82531 

1.21166 

28 

83 

.74131 

1.34896 

.76871 

.30087 

.79686 

.25492 

.82580 

1.21094 

27 

84 

.74176 

1.34814 

.76918 

.30009 

.79734 

.25417 

.82629 

1.21023 

26 

85 

.74221 

1.34732 

.76964 

.29931 

.79781 

.25343 

.82678 

1.20951  25 

86 

.74267 

1.34650 

.77010 

.29853 

.79829 

.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

.29775 

.79877 

.25193 

.82776 

1.20808 

23 

3s 

.74357 

1.34487 

.77103 

.29696 

.79924 

.25118 

.82825 

1.20736 

22 

3i» 

.74402 

1.34405 

.77149 

.29618 

.79972 

.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

.29541 

.80020 

.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

.29463 

.80067 

.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

.29385 

.80115 

.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

.29307 

.80163 

.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

.29229 

.80211 

.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

.29152 

.80258 

.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

.29074 

.80306 

.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

.28997 

.80354 

.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

.28919 

.80402 

.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

.28842 

.80450 

.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

.28764 

.80498 

.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

.28687 

.80546 

.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

.28610 

.80594 

.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

-  .28533 

.80642 

.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

.28456 

.80690 

.23931 

.83633 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

.28379 

.80738 

.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

.28302 

.80786 

.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

.28225 

.80834 

.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

.28148 

.80882 

.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

.28071 

.80930 

.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

.27994 

.80978 

.23490 

.83910 

1.19175 

0 

M. 

Cotang. 

Tang. 

Cotang.   Tang, 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

I      «• 

52° 

51° 

5O° 

TABLE  III.   NATUKAL  TANGENTS,  ETC. 


40° 

41° 

42° 

43° 

1 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M 

0 

.83910 

1.19175 

.86929 

1  .  15037 

.90040 

.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1  .  14969 

.90003 

.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.U902 

.90146 

.  10931 

.93360 

1.07112 

58 

3 

.84059 

1  .  18964 

.37082 

1.11834 

.90199 

.  1G867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

.  10802 

.93469 

1.06987 

ec 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

.  10737 

.93524 

1.06925 

55 

6 

.84208 

1  .  18754 

.87236 

1.14C32 

.90357 

.1067* 

.93578 

1.06862 

54 

7 

.84258 

1  .  18684 

.87287 

1.14565 

.90410 

.10607 

.93633 

1  .06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

9 

.84357 

1  .  18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1  .  18404 

.87492 

1.14296 

.90621 

1  .  10349 

.93852 

1.06551 

<L9 

12 

.84507 

1  .  18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

13 

.84556 

1.18264 

.87595 

1.14t62 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1  .  10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1  .  10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.909401  1.09963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

.1.09899 

.94235 

1.GG117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

1  .  13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1  .  13561 

.91206 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1  .  13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1,17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

J.  05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1  .  16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.95229 

1  .05010)  24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.95284 

1.04949  23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888  22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827  21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.047661  20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.047051  19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08389 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1.07990 

.95897 

1.04279 

12 

49 

.86368 

1.15783 

.89463 

1.11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1  .07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040 

1.11061 

•93252 

1.07237 

.96569 

1.03553 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

4<J" 

48° 

47  = 

46° 

TABLE  III.   NATURAL  TANGENTS,  ETC. 


44                      1 

44° 

44e 

M 

Tang.  1  Cotang.  M. 

M. 

Tang. 

Cotang. 

M. 

M.  |  Tang. 

Cotang. 

M. 

0 

.96569    1.03553     60 

20 

.97700 

1.02355 

40     40 

.98843 

.01170 

20 

1 

.96625 

1.03493     59 

21 

.97756 

1.022% 

39      41 

.98901 

.01112 

19 

2 

.96681 

1.03433     58 

22 

.97813 

1.02236 

38     42 

.98958 

.01053 

18 

i 

.96738 

1.03372     57 

23 

.97870 

1.02176 

37     43 

.99016 

.00994 

17 

4 

.96794 

1.03312     56 

24 

.97927 

1.02117 

36 

44 

.99073 

.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

.00876 

15 

G 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

46 

.99189 

.00818!  14 

•j 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

1.01879 

32 

.48 

.99304 

.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

.00642 

11 

to 

.97133 

1.02952 

50 

30 

.98270 

1.01761 

30 

50 

.99420 

.00583 

10 

11 

.97189 

1.02892     49 

31 

.98327 

1.01702 

29 

51 

.99478 

.00525 

9 

12 

.97246 

1.02832 

4* 

32 

.98384 

1.01642 

28 

52 

.99536 

.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

.00408 

7 

14 

.97359 

1.02713     46 

34 

.98499 

1.01524 

26 

54 

.99652 

.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

.00291 

5 

u 

.97472 

1.02593 

44 

36 

.98613 

1.01406 

24 

56 

.99768 

.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

M. 

Cotang 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M 

45' 

45* 

45° 

TABLE  IV. 


LOGAEITHMIC  SINES, 
TANGENTS, 


A2TD 


COTANGENTS. 


40        TABLE  IV.   LOGARITHMIC  SINES,  ETC. 
0-                                                 179" 

M. 

Sine. 

D.I". 

Cosine. 

D.  r 

Tang. 

D.  1". 

Cotang. 

M. 

0 

Inf.  neg. 

0.000000 

AA 

Inf.  neg. 

Infinite. 

60 

1 
2 
3 
4 
5 
6 
7 
8 
9 

6.463726 
.764756 
.940847 
7.065786 
.162696 
.241877 
.308824 
.366816 
.417968 

5017.17 
2934.85 
2082.31 
1615.17 
1319.68 
1115.75 
966.53 
852.54 
762.63 

.000000 
.000000 
.000000 
.000000 
.000000 
9.999999 
.999999 
.999999 
.999999 

.Utl 

.00 
.00 
.00 
.00 
.00 
.01 
.01 
.01 
.01 

6.463726 
.764756 
.940847 
7.065786 
.162696 
.241878 
.308825 
.366817 
.417970 

5017.17 
2934.83 
2082.31 
1615.17 
1319.69 
1115.78 
996.53 
852.54 
762.63 

13.536274 
.235244 
.059153 
12.934214 
.837304 
.758122 
.691175 
.633183 
.582030 

69 
58 
57 
56 
55 
54 
53 
52 
51 

10 

7.463726 

9.999998 

/\1 

7.463727 

12.536273 

50 

11 
12 
13 
14 
15 
16 
17 
18 

.505118 
.542906 
.577668 
.609853 
.639816 
.667845 
.694173 
.718997 

689  .  88 
629.81 
579.36 
536.41 
499.38 
467.14 
438.81 
413.72 

.999998 
.999997 
.999997 
.999996 
.999996 
.999995 
.999995 
9.999994 

.Ul 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

f\1 

.505120 
.542909 
.577672 
.609857 
.639820 
.667849 
.694179 
.719003 

629^81 
579.33 
536.42 
499.39 
467.15 
438.82 
413.73 

QO1  QfJ 

.494880 
.457091 
.422328 
.390143 
.360180 
.332151 
.305821 
.280997 

49 
48 
47 
46 
45 
44 
43 
42 

19 

.742477 

391  .  35 
371.27 

.999993 

.Ul 
.01 

.742484 

oy  i  .  oo 
371.28 

.257516 

41 

20 
21 
22 
23 
24 
25 
26 

28 
29 

7.764754 
.785943 
.806146 
.825451 
.843934 
.861662 
.878695 
.895085 
.910879 
.926119 

353.15 
336.72 
321.75 
308.05 
295.47 
283.88 
273.17 
263.23 
253.99 
245.38 

9.999993 
.999992 
.999991 
.999990 
.999989 
.999988 
.999988 
.999987 
.999986 
.999985 

.01 
.01 
.01 
.01 
.02 
.02 
.02 
.02 
.02 
.02 

7.764761 

.785951 
.806155 
.825460 
.843944 
.861674 
.878708 

.'910894 
.926134 

351.36 
336.73 
321.76 
308.06 
295.49 
283.90 
273.18 
263.25 
254.01 
245.40 

12.235239 
.214049 
.193845 
•  .174540 
.156056 
.138326 
.121292 

!089106 
.073866 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 

7.940842 
.955082 
.968870 
.982233 
.995198' 
8.007787 
.020021 
.031919 
.043501 

237.33 
229.80 
222.73 
216.08 
209.81 
203.90 
198.31 
193.02 

9.999983 
.999982 
.999981 
.999980 
.999979 
.999977 
.999976 
.999975 
.999973 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

7.940858 
.955100 
.968889 
.982253 
.995219 
8.007809 
.020045 
.031945 
.043527 

237.35 
229.81 
222.75 
216.10 
209.83 
203.92 
198.33 
193.05 

1  ftfi  AQ 

12.059142 
.044900 
.031111 
.017747 
.004781 
11.992191 
.979955 
.968055 
.956473 

30 

29 
28 
27 
26 
25 
24 
23 
22 

39 

.054781 

188.01 
183.25 

.999972 

!02 

.054809 

loo  .  Uo 

183.27 

.945191 

21 

40 
41 
42 
43 

8.065776 
.076500 
.086965 
.097183 

178.72 
174.41 
170.31 

9.999971 
.999969 
.999968 
.999966 

.02 
.02 

.02 

AO 

8.065806 
.076531 
.086997 
.097217 

178.74 
174.44 
170.34 

11.934194 
.923469 
.913003 
.902783 

20 
19 
18 
17 

44 
45 

.107167 
.116926 

166  .  39 
162.65 

.999964 
.999963 

.U^ 

.03 

/\Q 

.107202 
.116963 

162!  68 

1  F^Q  1A 

.892797 
.883037 

16 
15 

46 

.126471 

159.08 

.999961 

.Uo 

.126510 

ioy.  iu 

.873490 

14 

47 
48 
49 

.135810 
.144953 
.153907 

155.66 
152.38 
149.24 
146.22 

.999959 
.999958 
.999956 

.03 
.03 
.03 
.03 

.135851 
.144996 
.153952 

155.68 
152.41 
149.27 
146.27 

.864149 
.855004 
.846048 

13 
12 
11 

50 
51 
62 
63 
64 
65 
66 
67 
58 
69 
60 

8.162681 
.171280 
.179713 
.187985 
.196102 
.204070 
.211895 
.219581 
.227134 
.234557 
.241855 

143.33 
140.54 
137.86 
135.29 
132.80 
130.41 
128.10 
125.87 
123.72 
121.64 

9.999954 
.999952 
.999950 
.999948 
.999946 
.999944 
.999942 
.999940 
.999938 
.999936 
.999934 

.03 
.03 
.03 
.03 
.03 
.03 
.04 
.04 
.04 
.04 

8.162727 
.171328 
.179763 
.188036 
.196156 
.204126 
.211953 
.219641 
.227195 
.234621 
.241921 

143.36 
140.57 
137.90 
135.32 
132.84 
130.44 
128.14 
125.90 
123.76 
121.68 

11.837273 
.828672 
.820237 
.811964 
.803844 
.795874 
.788047 
.780359 
.772805 
.765379 
.758079 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

90° 


891 


TABLE   IV.      LOGARITHMIC  SINES,  ETC. 


41 

178C 


Sine. 


.241855 
.249033 
.256094 
.263042 


.276614 
.283243 


.296207 
.302546 

.308794 
.314954 
.321027 
.327016 
.332924 
.338753 
.344504 
.350181 
.355783 
.361315 

.366777 
.372171 
.377499 
.382762 
.387962 
.393101 


.408161 
.413068 

.417919 
.422717 
.427462 
.432156 
.436800 
.441394 
.445941 
.450440 
,454893 
,459301 


.467985 
.472263 

.476498 


.484848 


.497078 
.501080 

8.505045 
.508974 
.512867 
.516726 
.520551 


.528102 
.531828 
535523 


.542S19 


119.63 
117.68 
115.80 
113.98 
112.21 
110.50 
108.83 
107.21 
105.65 
104.13 

102.66 
101.22 
99.82 
98.47 
97.14 
95.86 
94.60 
93.38 
92.19 
91.03 

89.90 
88.80 
87.72 
86.67 
85.64 
84.64 


82.71 
81.77 
80.86 

79.96 
79.09 
78.23 
77.40 
76.57 
75.77 
74.99 
74.22 
73.46 
72.73 

72.00 
71.29 
70.60 


68.59 
67.94 
67.31 


65.48 
64.89 
64.31 
63.75 
63.19 
62.64 
62.11 
61.58 
61.06 
60.55 


Cosine. 


.999927 


.999918 
.999915 
.999913 

9.999910 
.999907 
.999905 


9.999882 


.999873 
.999870 


.999831 


.999786 
.999782 


.999774 


.999761 
.999757 
.999753 


.999744 
.999740 


M.    Cosine.     D.I",  j     Sine.     D.l".|  Cotang. 
91°  D 


D.1- 


Tang. 


8.241921 
.249102 
.256165 
.263115 


.315046 
.321122 
.327114 
.333025 


.344610 


.361430 


.372292 
.377622 


,888862 

.393234 
.398315 


.408304 
.413213 

8.418068 


.427618 
.432315 


.441560 
.446110 
.450613 
.455070 
.459481 

8.463849 
.468172 
.472454 
.476693 


.485050 
.489170 
.493250 
.497293 
.501298 

8.505267 
.509200 


.516961 
.520790 
.524586 


.5,32080 
.535779 
.539447 
.543084 


D.I".     Cotang.    M. 


119.67 
117.72 
115.84 
114.02 
112.25 
110.54 
108.87 
107.26 
105.70 
104.18 

102.70 
101.26 
99.87 
98.51 
97.19 
95.90 
94.65 
93.43 
92.24 
91.08 


87.77 
86.72 
85.70 
84.70 
83.71 
82.76 
81.82 
80.91 

80.02 
79.14 
78.30 
77.45 
76.63 
75.83 
75.05 
74.28 
73.52 
72.79 

72.06 
71.35 
70.66 


68.01 
67.38 
66.76 
66.15 

65.55 
64.96 
64.39 
63.82 
63.26 
62.72 
62.18 
61.65 
61.13 


L. 758079 
.750898 
.743835 
.736885 
.730044 
.723309 
.716677 
.710144 
.703708 
.697366 

11.691116 

.684954 
.678878 
.672886 
.666975 
.661144 
.655390 
.649711 
.644105 
.638570 


.GL'7708 
.622378 
.617111 
.611908 
.606766 
.601685 


.586787 

11.581932 
.577131 
.572382 
.567685 
.563038 
.558440 
.553890 
.549387 
.544930 
.540619 

11. 536151 
.531828 

.527546 
,523307 
519108 
,514950 
.510830 
.506750 
.502707 
.498702 

11.494733 

.490800 


.479210 
.475414 
.471651 
.467920 
.464221 
.460553 
.456916 


D.I".   Tang.   M. 


42 


TABLE   IV.      LOGARITHMIC   SINES,    ETC. 


2°                                                                                                                 177' 

M. 

Sine.       D.l". 

Cosine. 

D.l". 

Tang. 

D.l".  |  Cotung. 

M. 

0 

8.542819 

fift  04 

9.999735 

8.543084 

11.456916 

60 

1 
2 
3 
4 

.546422 
.549995 
.553539 
.557054 

DU.  vrr 

59.55 
59.06 

58.58 

KQ      *l  -t 

.999731 
.999726 
.999722 
.999717 

.07 
.07 
.08 

f\Q 

.546691 
.550268 
.553817 
.557336 

59^62 
59.14 
58.66 

.453309 
.449732 
.446183 
.442664 

59 
58 
57 
56 

5 

.560540 

Do  .  11 
57   fi5 

.999713 

•  Uo 
08 

.560828 

58.  19 

f\7   7^ 

.439172 

55 

6 

7 
8 

.563999 
.567431 
.570836 

O4  .  Dt> 

57.19 
56.74 

Kf     OA 

.999708 
.999704 
.999699 

.'08 
.08 

no 

.564291 
.567727 
.571137 

Dl  .  •  v 

57.27 
56.82 

.435709 
.432273 

.428863 

54 
53 

52 

9 

.574214 

oo.  ou 
65.87 

.999694 

•  Uo 

.08 

.574520 

55^95 

.425480 

61 

10 

8.577566 

55  44 

9.999689 

AQ 

8.577877 

KK     Kf) 

11.422123 

CO 

11 

.580892 

KK     fi9 

.999685 

.Uo 
fift 

.581208 

DO.  O^ 
err     -\f\ 

.418792 

.49 

12 

.584193 

OD.  \)& 

.999680 

•  Uo 

.584514 

Do.  1U 

.415486 

48 

13 

.587469 

M1O 

.999675 

*AQ 

.587795 

54.68 

KA      f)i-f 

.412205 

47 

14 
15 

.590721 
.593948 

.  1  J 

63.79 

pro    OQ 

.999670 
.999665 

.Uo 

.08 

/\Q 

.591051 
.594283 

D4  .  Zi 

53.87 

.408949 
.405717 

46 
45 

16 
17 

18 
19 

.597152 
.600332 
.603489 
.606623 

DO.  t>y 
53.00 
52.61 
52.23 
51.86. 

.999660 
.999655 
.999650 
.999645 

•  Uo 

.08 
.08 
.08 
.09 

.597492 
.600677 
.603839 
.606978 

53.47 
63.08 
52.70 
52.32 
51.94 

.402508 
.399323 
.396161 
.393022 

44 
43 

41 

20 

8.609734 

51  49 

9.999640 

8.610094 

11.389906 

40 

21 

.612823 

61  12 

.999635 

no 

.613189 

f-t  01 

.386811 

£9 

22 
23 
24 

25 
26 

.615891 
.618937 
.621962 
.624965 
.627948 

50!76 
50.41 
50.06 
49.72 

.999629 
.999624 
.999619 
.'999614 
.999608 

•Uy 
.03 
.09 
.03 
.03 

.616262 
.619313 
.622343 
.625352 
.628340 

ol  .^1 

60.85 
50.50 
60.15 
49.81 

.383738 
.380687 
.377657 
.374648 
.371660 

38 
37  - 
36 
35 

27 
28 

.630911 
.633854 

49!04 
48  71 

.999603 
.999597 

!09 

f)O 

.631308 
.634256 

49^13 

AQ     Qf\ 

.368692 
.365744 

£3 

32 

29 

.636776 

^to.  i  L 

48.39 

.999592 

.\j*7 

•  .09 

.637184 

4o.  oO 

48.48 

.362816 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

8.639680 
.642563 
.645428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 

48.06 
47.75 
47.43 
47.12 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 

9.999586 
.999581 
.999575 
.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 

.09 
.09 
.03 
.03 
.09 
.10 
.10 
.10 
.10 
.10 

8.640093 
.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.6626C9 
.6654S3 

48.16 
47.84 
47.53 
47.22 
46.91 
46.61 
46.31 
46.02 
45.73 
45.44 

11.359907 
.357018 
.354147 
.351296 
.348463 
.345848 
.342851 
.340072 
.337311 
.334567 

30 

29 

28 
27 

25 

23 

22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

8.667689 
.670393 
.673080 
.675751 
.678405 
.681043 
.683665 

!  688863 
.691438 

45.06 
44.79 
44.51 
44.24 
43.97 
43.70 
43.44 
43.18 
42.92 
42.67 

9.999529 
.999524 
.999518 
".999312 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 

.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 

8.6681CO 
.670870 
.673503 
.6762S9 
.678900 
.681544 
.684172 
.68G7F4 
.6893C1 
.691263 

45.16 

44.88 
44.61 
44.34 
44.07 
43.80 
43.54 
43.28 
43.03 
42.77 

11.331840 
.329120 
.326437 
.323761 
.321100 
.3184C6 
.315818 
.313216 
.310619 
.308037 

20 
» 
18 
17 
16  . 
15 
14 
13 
12 
11 

60 
61 

8.693998 
.696543 

42.42 

A.O    1  7 

9.9991C9 

.99D4G3 

.10 

8.094529 
.697081 

42.52 

11.305471 
.302919 

10 
9 

62 
63 
64 
65 
66 

.699073 
.701589 
.704090 
.706577 
.709049 

*±-  .  1  f 

41.92 
41.68 
41.44 
41.21 

Af\    Q7 

.9994C6 
.999450 
.999443 
.999437 
.999431 

.11 
.11 
.11 
.11 

.699617 
.7021  £9 
.704646 
707140 

42.^8 
42.  f  3 
41.79 
41.55 
41.32 

.300383 
.297861 
.295354 
.292860 
.290CS2 

8 
7 
6 
5 
4 

67 
68 
69 
60 

.711507 
.713952 
.716383 

.718800 

4v.y* 

40.74 
40.51 
40.29 

.999424 
.999418 
.999411 
.999404 

.11 
.11 
.11 

71208-5 
.714534 
.716972 
.719396 

41  .08 
40.85 
40.  62 
40.40 

.287917 
.285465 
.283028 
.280604 

3 
2 
1 
0 

M. 

Cosine. 

D.l    . 

Sine.     D.l  - 

Cotailg. 

D.l". 

92 


87^ 


TABLE  IV.      LOGARITHMIC   SINES,   ETC.  43 

176° 


Sine.      D.l.     Cosine.  |D. l.     Tang.      D.I".     Cotang.    M. 


8.718800 
.721204 
.723595 
.725972 
.728337 


.783027 

.735354 
.737667 


8.742259 
.744536 
.746802 


.751297 
.753528 
.755747 
.757955 
.760151 
.762337 

8.764511 
.766675 


.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
.783605 

8.785675 
.787736 
.789787 
.791828 
.793859 
.795881 
.797894 


.801892 


.807819 

.809777 
.811726 
.813667 
.815599 
.817522 


8.825130 
.827011 


.830749 
.832607 
.S34456 
.836297 
.838130 


.841774 


M.  I  Cosine. 
93° 


40.06 


39.41 
39.19 


38.77 
38.57 


38.16 

37.96 
37.76 
37.56 
37.37 
37.17 
36.98 
36.79 
36.61 
36.42 
36.24 

36.06 


35.70 
35.53 
35.35 
35.18 
35.01 
34.84 
34.67 
34.51 

34.31 
34.18 
34.02 


33.70 
33.54 
33.39 
33.23 
33.08 
32.93 

32.78 
32.63 
32.49 
32.34 
32.19 
32.05 
31.91 
31.77 
31.63 
31.49 

31.35 
31.22 
31.08 
30.95 


30.56 
30.43 
30.30 
30.17 


.999371 


.999315 


9.999265 
.999257 
.999250 


.999212 
.999205 
.999197 


.999181 
.999174 


.999150 
.999142 


9.999110 
.999102 


.999077 
.[99069 
.9S9061 
.999053 


0.999027 
.999019 


.999002 


.11 
.11 
.11 
.11 
.11 
.11 
.12 
.12 
.12 
.12 

.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 

.12 

.12 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

.13 
.13 
.14 
.14 
.14 
.14 
.14 
.14 
.11 
.14 

.14 
.14 

.14 
.14 
.14 
.14 
.14 
.15 
.15 
.15 


D  1".        Sine.    ID.i  '.   Cotang. 


.721806 
.724204 
.726588 


.731317 
.733663 
.735996 
.738317 
.740626 

8.742922 
.745207 
.747479 
.749740 
.751989 
.754227 
.756453 


.760872 
.763065 

8.765246 
.767417 
.769578 
.771727 
.773866 
.775995 
.778114 
.780222 


.784408 

8.786486 
.788554 
.790613 
.792662 
.794701 
.796731 
.798752 
.800763 
.802765 
.804758 

8.806742 

.808717 


.812641 
.814589 
.816T29 
.818461 
.820384 


.824205 

8.82C103 
.827992 
.829874 
.831748 


.835471 
.837321 
.8?9163 


.842*25 
.844644 


40.17 
39.95 
39.74 
39.52 
39.30 


38.48 
38.27 

38.07 
37  87 
37.68 
37.49 
37.29 
37.10 
36.92 
36.73 
36.  £5 


36.18 
36.00 
35.83 
35.65 
35.48 
35.31 
35.14 
34.97 
34.80 
34.64 

34.47 
34.31 
34.15 
33.99 
33.83 
33.68 
33.52 
33.37 
33.22 
33.07 

32.92 
32.78 
32.62 
32.48 
32.33 
32.19 
32.05 
31.91 
31.77 
31.63 

31.50 
31.36 
31.23 
31.10 


30.83 
30.70 
30.57 
bO.45 
30.32 


D.I". 


1.280604 
.278194 
.275796 
.273412 
.271041 


.264004 
.261683 
.259374 

1.257078 
.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239128 
.236935 

1.234754 
.232583 

!  228273 

.226134 
.224005 
.221886 
.219778 
.217680 
.215592 

11.213514 
.211446 
.208387 


.201248 

! 197236 

.195242 

11.193258 
.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 

11.173897 
.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 
.155356 


Tang. 


86C 


44 


TABLE   IV.      LOGARITHMIC  SINES,  ETC. 


175a 


M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

8.843585 
.845387 
-847183 
.848971 
.850751 
.852525 
.854291 
.856049 
.857801 
.859546 

30.05 
29.92 
29.80 
29.67 
29.55 
29.43 
29.31 
29.19 
29.07 
28.96 

9.993941 
.998932 
.993923 
.998914 
,99:905 
.993896 
.993887 
.993878 
.993869 
.993860 

.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 

8.844644 
.846455 
.848260 
.850057 
.851846 
-  .853628 
.855403 
.857171 
.858932 
.860686 

30.19 
30.07 
29.95 
29.82 
29.70 
29.58 
29.46 
29.35 
29.23 
29.11 

11.155356 
.153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 

8.861283 
.863014 
.864738 
.866455 
.868165 

28.84 
28.73 
28.61 
28.50 

9.998S51 
.998841 
.998832 
.998823 
.998813 

.15 
.15 
.15 
.16 

8.862433 
.864173 
.865906 
.867632 
.869351 

29.00 
28.88 
28.77 
28.66 

11.137567 
.135827 
.134094 
.132363 
.130649 

50 
49 
48 
47 
46 

15 
16 
17 
18 
19 

.869868 
.871565 
.873255 
.874938 
.876615 

28.28 
28.17 
28.06 
27.95 
27.86 

.998804 
.998795 
.998785 
.998776 
.998766 

.16 
.16 
.16 
.16 
.16 

.871064 
.872770 
.874469 
.876162 
.877849 

28.43 
28.32 
28.21 
28.11 
28.00 

.128936 
.127230 
.125531 
.123833 
.122151 

45 
44 
43 
42 
41 

20 

21 

8.878285 
.879949 

27.73 

9.998757 
.998747 

.16 

8.879529 
.881202 

27.89 

11.120471 
.118798 

40 
39 

22 
23 
24 
25 
26 
27 
28 
29 

.881607 
.883258 
.884903 
.886542 
.888174 
.889801 
.891421 
.893035 

27.52 
27.42 
27.31 
27.21 
27.11 
27.00 
26.90 
26.80 

.993738 
.998728 
.998718 
.998708 
.998690 
.998689 
.993679 
.998669 

.16 
.16 
.16 
.16 
.16, 
.16 
.16 
.17 

.882869 
.884530 
.886185 
.887833 
.889476 
.891112 
.892742 
.894365 

27.68 
27.58 
27.47 
27.37 
27.27 
27.17 
27.07 
26.97 

.117131 
.115470 
.113815 
.112167 
.110524 
.108883 
.107253 
.105634 

If 

36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 

8.894643 
.896246 
.897842 
.899432 
.901017 
.902596 

26.70 
26.60 
26.51 
26.41 
26.31 

9.998659 
.993649 
.998639 
.998629 
.998619 
.998609 

.17 
.17 
.17 
.17 
.17 

8.895984 
.897596 
.890203 
.900803 
.902398 
.903987 

26.87 
26.77 
26.67 
26.58 
26.48 

11.104016 
.102404 
.100797 
.099197 
.097602 
.096013 

30 
29 
28 
27 
26 
25 

36 
37 
38 

.904169 
.905736 
.907297 
.908853 

26.12 
26.03 
25.93 
25.84 

.993599 
.998589 
.998578 
.998568 

.17 
.17 
.17 
.17 

.905570 
.907147 
.908719 
.910285 

26.29 
26.20 
26.10 
26.01 

.094430 
.092853 
.091281 
.089715 

24 
23 
22 
21 

40 

41 
42 
43 
44 
45 
43 
47 
43 
49 

8.910404 
.911949 
.913488 
.915022 
.916550 
.918073 
.919591 
.921103 
.922610 
.924112 

25.75 
25.66 
25.56 
25.47 
25.38 
25.29 
25.20 
25.12 
25.03 
24.94 

9.998558 
.998548 
.998537 
.998527 
.998516 
.908506 
.998495 
.998485 
.993474 
.998464 

.17 
.17 
.17 
.17 
.18 
.18 
.18 
.18 
.18 
.18 

8.911846 
.913401 
.914951 
.916495 
.918034 
.919568 
.921096 
.922619 
.924136 
.925649 

25.92 
25.83 
25.74 
25.65 
26.56 
25.47 
25.38 
25.30 
25.21 
25.12 

11.088154 
.086599 
.085049 
.083505 
'  .081966 
.080432 
.078904 
.077381 
.075864 
.074351 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

59 
51 

8.925609 
.927100 

24.86 

9.998453 
.998442 

.18 

8.927156 
.928658 

25.03 

11.072844 
.071342 

10 
9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

.928587 
,930068 
.931544 
.933015 
.934481 
.935942 
.937398 
.938850 
.940296 

24.77 
24.69 
24.60 
24.52 
24.43 
24.35 
24.27 
24.19 
24.11 

.998431 
.998121 
.998410 
.998399 
.998388 
.998377 
.998366 
.998355 
.998344 

.18 
.18 
.18 
.18 
.18 
.18 
.18 
.18 
.18 

.930155 
.931647 
.933134 
.934616 
.936093 
.937565 
.939032 
.940494 
.941952 

24.95 
24.86 
24.78 
24.70 
24.61 
24.53 
24.45 
24.37 
24.30 

.069845 
.068353 
.066866 
.065384 
.063907 
.062435 
.060968 
.059506 
.058048 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Co^ino. 

D.I".   Pino,  n  i".  r-otang. 

D.I".   Tang. 

M. 

94C 


85 


TABLE  IV.      LOGARITHMIC  SINES,  ETC.             46 

5"                                                                                                                1748 

M. 

Sine. 

D.l". 

Cosine. 

D.I  V 

Tang. 

D.I". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 
6 
7 
8 
9 

8.940296 
1941738 
.943174 
.944606 
.946034 
.947456 
.948874 
.950287 
.951696 
.953100 

24.03 
23.94 
23.87 
23.79 
23.71 
23.63 
23.55 
23.48 
23.40 
23.32 

9.998344 
.998333 
.998322 
.998311 
.998300 
.998289 
.998277 
.998266 
.998255 
.998243 

.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 

8.941952 
.943404 
.944852 
.946295 
.947734 
.949168 
.950597 
.952021 
.953441 
.954856 

24.21 
24.13 
24.05 
23.97 
23.90 
23.82 
23.74 
23.67 
23.60 
23.51 

11.058048 
.056596 
.055148 
.053705 
.052266 
.050832 
.049403 
.047979 
.0-16559 
.045144 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 

8.954499 
.955894 
.957284 

23.25 
23.17 

9.998232 
.998220 
.998209 

.19 

.19 

8.956267 
.957674 
.959075 

23.44 
23.37 

11.043733 

.042326 
.040925 

50 
49 

48 

13 
14 
15 
16 
17 
18 
19 

.958670 
.960052 
.961429 
.S62801 
.964170 
.C65534 
.166893 

23.10 
23.02 
22.95 
22.88 
22.80 
22.73 
22.66 
22.59 

.998197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998128 

.19 
.19 
.19 
.19 
.19 
.19 
.20 
.20 

.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 

23.29 
23.22 
23.14 
23.07 
23.00 
22.93 
22.86 
22.79 

.039537 
.038134 
.036745 
.035361 
.033981 
.032606 
.031234 

47 
46 
45 
44 
43 
42 
41 

20 

8.968249 

9.998116 

8.970133 

11.029867     40 

21 

.969600 

22.52 

.998104 

.20 

.971496 

22.72 

.028501     39 

22 

.970947 
.972289 

22.45 
22.38 

.998092 
.998080 

.20 
.20 

OA 

.972855 
.974209 

22.65 
22.57 

077145 

38 
37 

24 
26 
26 
27 
28 

.973628 
.974962 
.976293 
.977619 
.978941 

22.31 
22.24 
22.17 
22.10 
22.03 

.998068 
.998056 
.998044 
.998032 
.998020 

.M 
.20 
.20 
.20 
.20 

.975560 
.976906 
.978248 
.979586 
.980921 

22.51 
22.44 
22.37 
22.30 
22.23 

.0:)4440 
02309* 
.021753 
.020414 
.019079 

36 
35 
34 
33 
32 

29 

.£'80259 

21.97 
21.90 

.998008 

.20 
.20 

.982251 

22.17 
22.10 

.017749 

31 

30 
31 

8.981573 

.982883 

21.83 

9.9979S6 
.997984 

.20 

8.983577 
.984899 

22.04 

11.016423 
.015101 

30 

29 

32 
33 
34 

35 
36 
37 
38 
39 

.984189 
.985491 
.986789 
.988083 
.989374 
.990660 
.891943 
.993222 

21  .77 
21.70 
21.63 
21.57 
21.50 
21.44 
21.38 
21.31 
21.25 

.997972 
.997959 
.997947 
.997935 
.997922 
.997910 
.997897 
.997885 

.20 
.20 
.20 
.20 
.21 
.21 
.21 
.21 
.21 

.986217 
.987532 
.988842 
.990149 
.991451 
.992750 
.994045 
.995337 

21.97 
21.91 
21.84 
21.78 
21.71 
21.65 
21.58 
21.52 
21.46 

.013783 
.012468 
.011158 
.009851 
.C08549 
.007250 
.005955 
.004663 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 

8.994497 
.995768 
.997036 
,998299 
.999560 
9.000816 
.C02069 

21.19 
21.12 
21.06 
21.00 
20.94 
20.88 

9.997872 
.997860 
.997847 
.997835 
.997822 
.997809 
.S97797 

.21 
.21 
.21 
.21 
.21 
.21 

8.996624 
.997908 
.999188 
9.C00465 
.C01738 
.C03007 
.C04272 

21.40 
21.34 
21.27 
21.21 
21.15 
21.09 

11.003376 
.002092 
.OC0812 
10  999535 
•I9S263 
9nti093 
.91)5728 

20 
19 
18 
17 
16 
15 
14 

47 
48 
49 

.003318 
.004563 
.005805 

20^76 
20.70 
20.64 

.997784 
.997771 

.997758 

.21 
.21 
.21 
.21 

.CC5534 
.C06792 
.008047 

21.03 
20.97 
20.91 
20.86 

.994466 
.993208 
.991953 

13 
12 
11 

50 
51 
52 
53 
54 
55 
66 
57 
58 
59 
60 

9.007044 
.008278 
.009510 
.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 
.019235 

20.58 
20.52 
20.46 
20.40 
20.34 
20.29 
20.23 
20.17 
20.12 
20.46 

9.997745 
.997732 
.997719 
.997706 
.997693 
.997680 
.997667 
.997654 
.997641 
.997628 
.997614 

.21 
.21 
.21 
.21 

.22 
.22 
.22 
.22 
.22 
.22 

9.009298 
.010546 
.011790 
.013031 
.014268 
.015502 
.016732 
.017959 
.019183 
.020403 
.021320 

20.80 
20.74 
20.68 
20.62 
20.56 
20.51 
20.45 
20.40 
20.33 
20.28 

10.990702 
.989154 
.988210 
.986969 
.985732 
.984498 
.983268 
.982041 
.980817 
.979C97 
.978380 

10 
9 
8 
7 
6 
5 
4 
3 

1 

0 

M. 

Cosine, 

D.r'.       Sine. 

Drl". 

Cotang. 

D.I". 

Tang. 

liT 

95'                                                                                                              84° 

46             TABLE  IV.      LOGARITHMIC  SINES,  ETC. 

6"                                                                                                                           173° 

M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 

9.019235 
.020435 
.021632 
.022825 
.024016 
.025203 
.026386 

20.00 
19.95 
19.89 
19.84 
19.78 
19.73 

1Q   &7 

9.997614 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 

.22 
.22 
.22 
.22 
.22 
.22 

9.021620 
.022834 
.024044 
.025251 
.026455 
.027655 
.028852 

20.23 
20.17 
20.12 
20.06 
20.01 
19.95 

.0.978380 
.977166 
.975956 
.974749 
.973545 
.972345 
.971148 

61 
5J 
53 
57 
53 
53 
51 

7 

.027567 

ly.Di 

.997520 

,23 

.030046 

19.90 

.969954 

53 

8 

.028744 

19.62 

.997507 

.23 

.031237 

19.85 

.968763 

52 

9 

.029918 

19.57 
19.52 

.997493 

.23 
.23 

.032425 

19.79 
19.74 

.967575 

51 

10 

9.031089 

9.997480 

9,033609 

10.966391 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

.032257 
.033421 
.034582 
.035741 
.036896 
.038048 
.039197 
.040342 
.041485 

19!41 
19.36 
19.30 
19.25 
19.20 
19.15 
19.10 
19.05 
19.00 

.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 

°23 
.23 
.23 
.23 
.23 
.23 
.23 
.23 
.23 

.034791 
o  035969 
.037144 
.  03831  6 
.039485 
.040651 
.0418"13 
.042973 
.044130 

19.  '64 
19.58 
19.53 
19.48 
19.43 
19.38 
19.33 
19.28 
19.23 

.965209 
.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 

49 
'48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 

9.042625 
.043762 
.044895 

18.95 
18.90 

1ft    QK 

9.997341 
.997327 
.997313 

.23 
.24 

()A 

9.045284 
.046434 

.047582 

19.18 
19.13 

10.954716 
.953566 
.952418 

40 
39 
38 

23 

.046026 

-lu.oO 
1Q    QA 

.997299 

.zA 

.048727 

19.08 

.951273 

37 

24 
25 

.047154 
.048279 

Jo.oU 

18.75 

.997285 
.997271 

.24 

.24 

.049869 
.051008. 

19.03 

18.98 

.950131 

.948992 

36 
35 

26 

.049400 

1  Q    £K 

.997257 

.24 

.052144 

18.93 

.947856 

34 

27 
28 
29 

.050519 
.051635 
.052749 

lo.oo 
18.60 
18.55 
18.50 

.997242 
.997228 
.997214 

.24 
.24 
.24 
.24 

.053277 
.054407 
,055535 

18.89 
18.84 
18.79 
18.74 

.946723 
.945593 
.944465 

33 
32 
31 

30 
31 
32 
33 
34 
35 

9.053859 
.054966 
.056071 
.057172 
.058271 
.059367 

18.46 
18.41 
18.36 
18.31 

18.27 

1  Q     OO 

9.997199 
.997185 
.997170 
.997156 
.997141 
.997127 

.24 
.24 
.24 
.24 
.24 

9.056659 
.057781 
.058900 
.060016 
.061130 
.062240 

18.70 

18.65 
18.60 
18.56 
18.51 

10.943341 
.942219 
.941100 
.939984 
.938870 
.937760 

30 
29 

28 
27 
26 
25 

36 

37 
38 
39 

.060460 
.061551 
.062639 
.063724 

lo.  ZiL 
18.17 
18.13 
18.08 
18.04 

.997112 
,997098 
.997083 
o  997068 

^24 
.24 
.25 

.25 

.063348 
.064453 
.065556 
.066655 

18.46 
18.42 
18.37 
18.33 
18.28 

.936652 
.935547 
.934444 
.933345 

24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.064806 
.065885 
.066962 
.068036 
.069107 
.070176 
.071242 
.072306 
.073366 
.074424 

17.99 
17.95 
17.90 
17.86 
•  17.81 
17.77 
17.72 
17.68 
17.64 
17.59 

9.997053 
.997039 
.997024 
.997009 
.996994 
.996979 
o  996964 
.996949 
.996934 
.996919 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
,25 
,25 
.25 

9.067752 
.068846 
.069938 
.071027 
.072113 
.073197 
.074278 
.075356 
.076432 
.077505 

18.24 
18.19 
18.15 
18.10 
18.06 
18.02 
17.97 
17.93 
17.89 
17.84 

10.932248 
.931154 
.930062 
.928973 
.927887 
.926803 
.925722 
.924644 
.923568 
.922495 

20 
19 
18 
17 
16 
13 
14 
13 
1  2 

11 

50 
51 
52 
53 
54 
55 
56 
57 
58 

9.075480 
.076533 
.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 

17.55 
17.51 
17.46 
17.42 
17.38 
17.34 
17.29 
17.25 
17  21 

9.996904 
.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 

.25 
.25 
.25 
.25 
.25 
.25 
.26 
.26 

9.078576 
.  079644 
.080710 
.081773 
.082833 
.083891 
.084947 
.086000 
.087050 

17.80 
17.76 
17.72 
17.67 
17.63 
17.59 
17.55 

17.51 
nftf 

10.921424 
.920356 
.919290 
.918227 
.917167 
.916109 
.915053 
,914000 
.912950 

10 
9 
8 
7 
6 
5 
4 
3 
2 

59 
60 

.084864 
.085894 

17  '.17 

.996766 
.996751 

!26 

.088098 
.089144 

.47 
17.43 

.911902 
.910856 

1 
0 

1*7 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 

36° 


TABLE  IV.      LOGARITHMIC  SINES,   ETC.  47 

172° 


M. 

Sine.' 

D.I".     Cosine.  |D.  1        Tang.      D.I".     Cotang.    M. 

0 
1 
2 
3 
4 

9.085894 
.086922 
.087947 
.088970 
.089990 

17.13 
17.09 
17.05 
17.00 

9.996751 
.996735 
.996720 
.996704 

.26 

.26 
.26 
.26 

9.089144 
.090187 
.091228 
.092266 
.093302 

17.39 
17.35 
17.31 
17.27 

10.910856 
.909813 
.908772 
.907734 
.906698 

60 
59 

58 
57 
FO 

5 
6 

.091008 
.092024 

16.96 
16.92 

.996673 
.996657 

.26 
.26 

.094336 
.095367 

17.19 

.905664 
.904633 

55 
54 

7 
8 
9 

.093037 
.094047 
.095056 

16.84 
16.80 
16.76 

.996641 
.996625 
.996610 

.26 
.26 
.26 

.096395 
.097422 
.098446 

17.11 
17.07 
17.03 

.903605 
.902578 
.901554 

53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

9.096062 
.097065 
.098066 
.099065 
.100062 
.101056 
.102048 
.103037 
.104025 
.105010 

16.73 
16-68 
16.65 
16.61 
16.57 
16.53 
16.49 
16.46 
16.43 
16.38 

9.996594 
.996578 
.996562 
.996546 
.996530 
.996514 
.996498 
.996482 
.996465 
.996449 

.26 

.27 
.27 
.27 
.27 
.27 
•27 
.27 
.27 
.27 

9.099468 
.100487 
.101504 
.102519 
.103532 
.104542 
.105550 
.106556 
.107559 
.108560 

16.99 
16.95 
16.91 
16.88 
16.84 
16.80 
16.76 
16.72 
16.69 
16.65 

10.900532 
.899513 
.898496 
.897481 
.896468 
.895458 
.894450 
.893444 
.892441 
.891440 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.105992 
.106973 
.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 

16.34 
16.30 
16.27 
16.23 
16.19 
16.16 
16.12 
16.08 
16.05 
16.01 

9.996433 
.996417 
.996400 
.996384 
.996368 
.996351 
.996335 
.996318 
.996302 
.996285 

.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.28 
.28 

9.109559 
.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 
.118452 

16.61 
16.58 
16.54 
16.50 
16.47 
16.43 
16.39 
16.36 
16.32 
16.29 

10.890441 
.889444 
.888449 
.887457 
!  886467 
.885479 
.884493 
.883509 
.882528 
.881548 

40 
39 
33 
37 
30 
35 
S4 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 

9.115698 
.116656 
.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 

15.98 
15.94 
15.90 
15.87 
15.83 
15.80 
15.76 
15.73 

9.996269 
.996252 
.996235 
.996219 
.996202 
.996185 
.996168 
.996151 
.996134 

.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 

9.119429 
.120404 
.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 

16.25 
16.22 
16.18 
16.15 
16.11 
16.08 
16.04 
16.01 

10.880571 
.879596 
.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872828 

30 
29 
28 

2G 
25 
24 
23 
22 

39 

.124248 

15.66 

.996117 

.28 

.128130 

15.94 

.871870 

21 

40 
41 
42 
43 
44 

9.125187 
.126125 
.127060 
.127993 
.128925 

15.62 
15.59 
15.56 
15.52 

9.996100 
.996083 
.996066 
.996049 
.996032 

.29 
..29 
.29 
.29 

9Q 

9.129087 
.130041 
.130994 
.131944 
.132893 

15.91 
15.87 
15.84 
15.81 

I*     *7 

10.870913 
.869959 
.869006 
.868056 
.867107 

20 
ID 
18 
17 
16 

45 
46 
47 
48 
49 

.129854 
.130781 
.131706 
.132630 
.133551 

15.45 
15.42 
15.39 
15.35 
15.32 

.996015 
.995998 
.995980 
.995963 
.995946 

.29 
.29 
.29 
.29 
.29 

.133839 
.134784 
.135726 
.136667 
.137605 

15.74 
15.71 
15.68 
15.64 
15.61 

.866161 
.865216 
.864274 
.863333 
.862395 

15 
14 
13 
12 
11 

50 

9.134470 

15.29 

9.995928 

.29 

9.138542 

15.58 

10.861458 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

.136303 
.137216 
.138128 
.139037 
.139944 
.140850 
.141754 
.142655 
.143555 

15.26 
15.22 
15.19 
15.16 
15.13 
15.09 
15.06 
15.03 
15.00 

.995894 
.995876 
.995859 
.995841 
.995823 
.995806 
.995788 
.995771 
.995753 

.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 

.140409 
.141340 
.142269 
.143196 
.144121 
.145044 
.145966 
.146885 
.147803 

15.55 
15.51 
15.48 
15.45 
15.42 
15.39 
15.36 
15.32 
15.29 

.859591 
.858660 
.857731 
.856804 
.855879 
.854956 
.854034 
.853115 
.852197 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I    . 

rotang 

D.  1". 

Tang. 

M. 

97° 


48  TABLE   IY.      LOGARITHMIC  SINES,   ETC. 

8° 


171' 


M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

9.143555 

-f  A  AT 

9.995753 

QA 

9.147803 

1*>  9R 

10.852197 

60 

1 

.144453 

14.y  i 

.995735 

.oU 

QA 

.148718 

IO.  Z\y 

•IK  OO 

.851282 

59 

2 
3 

.145349 
.146243 

14  .'90 

UQ7 

.995717 
.995699 

•  oU 

.30 

.149632 
.150544 

10.  £o 
15.20 

•IK  17 

.850368 
.849456 

58 
57 

4 
5 
6 

.147136 
.148026 
.148915 

.  O* 

14.84 
14.81 

1/1  7Q 

.995681 
.995664 
.995646 

!30 
.30 

Q/\ 

.151454 
.152363 
.153269 

10.  1  1 

15.14 
15.11 

-IK  AQ 

.848546 
.847637 
.846731 

56 
55 
54 

7 
8 
9 

.149802 
.150686 
,151569 

14.  *O 

14.75 
14.72 
14.69 

.995628 
.995610 
.995591 

•  oU 

.30 
.30 
.30 

.154174 
.155077 
.155978 

10.  Uo 

15.05 
15.02 
14.99 

.845826 
.  844923 
.844022 

53 
52 
51 

io 

9.152451 

1,4  fid 

9.995573 

QA 

9.156877 

14  Qfi 

10.843123 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

.153330 
.154208 
.155083 
.155957 
.156830 
.157700 
.158569 
.159435 
.160301 

14  •OU 

14.63 
14.60 
14.57 
14.54 
14.51 
14.48 
14.45 
14.42 
14.39 

.995555 
.995537 
.995519 
.995501 
.995482 
.995464 
.995446 
.995427 
.995409 

.oU 

.30 
.30 
.30 
.31 
.31 
.31 
.31 
.31 
.31 

.157775 
.158671 
.159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 

14!  93 
14.90 
14.87 
14.84 
14.81 
14.78 
14.75 
14.73 
14.70 

.842225 
.841329 
.840435 
.839543 
.838653 
.837764 
.836877 
.835992 
.835108 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

9.1611G4 

9.995390 

91 

9.165774 

10.834226 

40 

21 
22 
23 

.162025 
.162885 
.163743 

14.36 
14.33 
14.30 

1/4  O7 

.993372 
.993353 
.995331 

•  ol 

.31 
.31 

91 

.166654 
.167532 
.168409 

14.67 
14.64 
14.61 

1/4  FCC 

.833346 
.832468 
.831591 

33 
33 
37 

24 
25 
26 

27 
28 

.164600 
.165454 
.166307 
.167159 
.168008 

14  .  Zi 

14.24 
14.22 
14.19 
14.16 

.995316 
.995297 
.995278 
.995260 
.995241 

•  ol 
.31 
.31 
.31 
.31 

.169284 
.170157 
.171029 
.171899 
.172767 

14.  5o 

14.56 
14.53 
14.50 
14.47 

.830716 
.829843 
.828971 
.828101 
.827233 

36 
35 

:u 

33 
32 

29 

.168856 

14.13 
14.10 

.995222 

.32 

.32 

.173634 

14.44 
14.42 

.826366 

31 

30 
31 

9.169702 
.170547 

14.07 

-1  4  APi 

9.995203 
.995184 

.32 

OO 

9.174499 
.175362 

14.39 

1/1  Q£ 

10.825501 
.824638 

30 
29 

32 

.171389 

14.  Uo 

.995165 

cOufi 

.176224 

14.OD 

.823776 

28 

33 
34 

.172230 
.173070 

14.02 
13.99 

.995146 
.995127 

'.32 

.177084 
.177942 

14.33 
14.31 

.822916 
.822058 

27 
26 

35 
36 
37 

.173908 
.174744 
.175578 

13.96 
13.94 
13.91 

.995108 
.995089 
.995070 

'.32 
.32 

OO 

.178799 
.179655 
.180508 

14.28 
14.25 
14.23 

.821201 
.820345 
.819492 

25 
24 
23 

38 
39 

.176411 
.177242 

13.88 
13.85 
13.83 

.995051 
.995032 

.0* 

.32 
.32 

.181360 
.182211 

14.20 
14.17 
14.15 

.818640 
.817789 

22 
21 

40 
41 
42 
43 
44 

9.178072 
.178900 
.179726 
.180551 
.181374 

13.80 
13.77 
13.75 
13.72 

9.995013 
.994993 
.994974 
.994955 
.994935 

.32 
.32 
.32 
.32 

9.183059 
.183907 
.184752 
.185597 
.186439 

14.12 
14.09 
14.07 
14.04 

10.816941 
.816093 
.815248 
.814403 
.813561 

20 
19 
18 
17 
16 

45 
46 

47 
48 
49 

.182190 
.183016 
.183834 
.184651 
.185466 

13.69 
13.67 
13.64 
13.61 
13.59 
13.56 

.994916 
.994896 
.994877 
.994857 
.994838 

.32 
.33 
.33 
.33 
.33 
.33 

.187280 
.188120 
.188958 
.189794 
.190629 

14.02 
13.99 
13.97 
13.94 
13.91 
13.89 

.812720 
.811880 
.811042 
.810206 
.809371 

15 
14 
13 
12 
11 

50 
51 
52 
53 

9.186280 
.187092 
.187903 
.188712 

13.54 
13.51 
13.48 

9.994818 
.994798 
.994779 
.994759 

.33 
.33 
.33 

9.191462 
.192294 
.193124 
.193953 

13.86 
13.84 
13.81 

10.808538 
.807706 
.806876 
.806047 

10 
9 
8 

7 

54 
55 
56 

57 

.189519 
.190325 
.191130 
.191933 

13.46 
13.43 
13.41 
13.38 

f  O  nf* 

.994739 
.994719 
.994700 
.994680 

.33 
.33 
.33 
.33 

OQ 

.194780 
.195606 
.196430 
.197253 

13.79 
13.76 
13.74 
13.71 

.805220 
.804394 
.803570 
.802747 

6 
5 
4 
3 

68 
59 
60 

.192734 
.193534 
.194332 

16.  OO 

13.33 
13.31 

.994660 
.994640 
.994620 

.DO 

.33 

.33 

.198074 
.198894 
.199713 

13!  66 
13.64 

.801926 
.801106 
.800287 

2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

81* 


TABLE   IV.      LOGARITHMIC  SINES,   ETC.  49 

170= 


M. 

Sine. 

D.  1". 

Cosine. 

D.I  . 

Tang. 

D.  1". 

Cotang. 

M- 

0 

1 

2 
3 
4 
5 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 

1  <\<U  i'~H 

13.28 
13.26 
13.23 
13.21 
13.18 
13.16 

9.994620 
.994600 
.994580 
.994560 
.994540 
.994519 

OOU.QO 

.33 
.33 
.33 
.34 
.34 
.34 

9.199713 

.200529 
.201345 
.202159 
.202971 
.203782 

0/UKQ9 

13.62 
13.59 
13.57 
13.54 
13.52 
13.49 

10.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795408 

^50~ 
59 
58 
67 
56 
65 
64 

7 
8 
9 

.  jyyuyi 
.199879 
.20066G 
.201451 

13.13 
13.11 
13.08 
13.06 

.  JcM-tyj 

.994479 
.994459 
.994438 

.34 
.34 
.34 
.34 

.zirioy^ 
.205400 
.206207 
.207013 

13.47 
13.45 
13.42 
13.40 

!  794600 
.793793 
.792987 

53 
C2 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.202234 
.203017 
.203797 
.204577, 
.205354 
.206131 
.206906 
.207679 
.208452 
.209222 

13.04 
13.01 
12.99 
12.96 
12.94 
12.92 
12.89 
12.87 
12.85 
12.82 

9.994418 
.994398 
.994377 
.994357 
.994336 
.994316 
.994295 
.994274 
.994254 
.994233 

.34 
.34 
.34 
.34 
.34 
.34 
.34 
.35 
.35 
.35 

9.207817 
.208619 
.209420 
.210220 
.211018 
.211815 
.212611 
.213405 
.214198 
.214989 

13.38 
13.35 
13.33 
13.31 
13.28 
13.26 
13.24 
13.21 
13.19 
13.17 

10.792183 

.791381 
.790580 
.789780 
.788982 
.788185 
.787389 
.786595 
.785802 
.785011 

CO 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.209992 
.210760 
.211526 
.212291 
.213055 
.213818 
.214579 
.215338 
210097 
.216854 

12.80 
12.78 
12.75 
12.73 
12.  71 
12.68 
12.66 
12.64 
12.62 
12.59 

9.994212 
.994191 
.994171 
.994150 
.994129 
.994108 
.994087 
.994066 
.994045 
.994024 

.35 

.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

9.215780 
.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 

13.15 
13.12 
13.10 
13.08 
13.06 
13.03 
13.01 
12.99 
12.97 
12.95 

10.784220 
.783432 
.782644 
.781858 
.781074 
.780290 
.779508 
.778728 
.777948 
.777170 

40 
39 
38 
3T 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 

9.217609 
.218363 

.219110 
.210868 
.220618 

12.57 
12.55 
12.53 
12.50 
19  dR 

9.994003 
.993982 
.993960 
.993939 
.993918 

.35 

.35 
.35 
.35 

0/5 

9.223607 
.224382 
.225156 
426929 

.226700 

12.92 
12.90 
12.88 
12.86 

19  A4 

10.776393 

.775618 
.774844 
.774071 
.773300 

30 
29 

28 
27 
26 

35 
36 
37 
38 
39 

.221367 
.222115 
.222861 
.223606 
.224349 

1.4.  4o 

12.46 
12.44 
12.42 
12.39 
12.37 

.993897 
.993875 
.993854 
.993832 
.993811 

.OO 

.36 
.36 
.36 
.36 
.36 

.227471 

.22823'' 
.229007 
.229773 
.230539 

1.4.CTZ 

12.82 
12.79 
12.77 
12.75 
12.73 

.772529 
.771761 
.770993 
.770227 
.769461 

26 
24 
23 
22 
21 

40 
41 

9.225092 
.225833 

12.35 
10  oo 

9.993789 
.993768 

.36 

9.231302 

.232065 

12.71 

10.768698 
.767935 

20 

19 

42 
43 
44 

.226573 
.227311 

.228048 

la.&f 

12.31 
12.29 

19     'V 

.993746 
.993725 
.993703 

-.36 
.36 
.36 

0/» 

.232826 
.233586 
.234345 

12.69 
12.67 
12.65 

•f  o    /»*> 

.767174 
.766414 
.765655 

18 
17 
16 

45 
46 
47 

48 
49 

.228784 
.229518 
.230252 
.230984 
.231715 

Mj6mlBv 

12.24 
12.22 
12.20 
12.18 
12.16 

.993681 
.993660 
.993638 
.993616 
.993594 

.00 
.36 
.36 
.36 
.36 
.36 

.235103 
.235859 
.236614 
.237368 
.238120 

IZ.Oo 

12.60 
12.58 
12.56 
12.54 
12.52 

.764897 
.764141 
.763386 
.762632 
.761880 

15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
67 

9.232444 
.233172 
.233899 
.234625 
.235349 
.236073 
.236795 
.237515 

12.14 
12.12 
12.10 
12.07 
12.05 
J2.03 
12.01 

1  1    OQ 

9.993572 
.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 

.36 

.37 
.37 
.37 
.37 
.37 
.37 

9.238872 
.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 

12.50 
12.48 
12.46 
12.44 
12.42 
12.40 
12.38 

1O    Q£ 

10.761128 
,760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 

10 
9 
8 
7 
-6 
6 
4 
3 

58 
59 
60 

.238235 
.238953 
.239670 

11  .  yy 
11.97 
11.95 

.993396 
.993374 
.993351 

.37 
.37 
.37 

.244839 
.245579 
.246319 

U.dO 

12.34 
12.32 

.755161 
.754421 
.753681 

2 
1 

0 

M. 

Cosine. 

D.r. 

Sine. 

D.I" 

Cotang. 

D.  1". 

Tang. 

M 

99°                                                                                                                   80° 

50             TABLE  IV.      LOGARITHMIC  SINES,   ETC. 

1O'                                                                                                                        169° 

M. 

Sine. 

D.I". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 
1 

2 
3 
4 
5 
6 

9.239670 
.240386 
.241101 
.241814 
.242526 
.243237 
.243947 

11.93 
11.91 
11.89 
11  .37 
11.85 
11.83 

noi 

9.993351 
.993329 
.993307 
.993284 
.993262 
.993240 
.993217 

.37 
.37 
.37 
.37 
.37 
.37 

OQ 

9.246319 
.247057 
.247794 
.248530 
.249264 
.249998 
.250730 

12.30 
12.28 
12.26 
12.24 
12.22 
12.20 

10.753681 
.752943 
.752206 
.751470 
.750736 
.750002 
.749270 

60 
59 
58 
57 
56 
55 
54 

7 
8 

.244656 
.245363 

.OJL 

11.79 

nil 

.993195 
.993172 

.OO 

.38 

OQ 

.251461 
.252191 

12.18 
12.17 

.748539 
.747809 

53 

52 

9 

.246069 

..<• 
11.75 

.993149 

.OO 

.38 

.252920 

12.15 
12.13 

.747080 

61 

10 
11 

9.246775 

.247478 

11.73 

n71 

9.993127 
.993104 

.38 

00 

9.253648 
.254374 

12.11 

10.746352 
.745626 

50 
49 

12 

.248181 

•  k  \. 

11  69 

.993081 

.00 
OQ 

.255100 

12.09 

.744900 

48 

13 
14 
15 
16 
17 
18 
19 

.248883 
.249583 
.250282 
.250980 
.251677 
.252373 
.253067 

ll!67 
11.65 
11.63 
11.61 
11.59 
11.58 
11.56 

.993059 
.993036 
.993013 
.992990 
.992967 
.992944 
.992921 

•  OO 

.38 
.38 
.38 
.38 
.38 
.38 
.38 

.255824 
.256547 
.257269 
.257990 
.258710 
.259429 
.260146 

12l  05 
12.03 
12.01 
12.00 
11.98 
11.96 
11.94 

.744176 
.743453 
.742731 
.742010 
.741290 
.740571 
.739854 

47 
46 
45 
44 
43 
42 
41 

20 
21 
22 

9.253761 
.254453 
.255144 

11.54 
11.52 
11  50 

9.992898 
.992875 
.992852 

.38 
.38 

OQ 

9.260863 
.261578 
.262292 

11.92 
11.90 

nQO 

10.739137 
.738422 
.737708 

40 

39 

38 

23 
24 

.255834 
.256523 

ll!48 

nAf* 

.992829 
.992806 

•  OO 

.39 

.263005 
.263717 

.89 
11.87 

.736995 
.736283 

37 
36 

25 
26 
27 
28 

.257211 

.257898 
.258583 
•   .259268 

•VO 

11.44 
11.42 
11.41 
11  3Q 

.992783 
.992759 
.992736 
.992713 

.39 
.39 
.39 
.39 

'Ml 

.264428 
.265138 
.265847 
.266555 

11.85 
11.83 
11.81 
11.79 

.735572 
.734862 
.734153 
.733445 

35 

34 
33 
32 

29 

.259951 

11  .'37 

.992690 

.o9 
.39 

.267261 

11.78 
11.76 

.732739 

31 

30 
81 
32 
33 

9.260633 
.261314 
.261994 
.262673 

11.35 
11.33 
11.31 
11  30 

9.992666 
.992643 
.992619 
.992596 

.39 
.39 
.39 

9.267967 
.268671 
.269375 
.270077 

11.74 
11.72 
11.70 

10.732033 
.731329 
.730625 
.729923 

30 
29 
28 
27 

34 
35 
36 
37 

38 
39 

.263351 
.264027 
.264703 
.265377 
.266051 
.266723 

1L28 
11.26 
11.24 
11.22 
11.20 
11.19 

.992572 
.992549 
.992525 
.992501 
.992478 
.992454 

'.39 
.39 
.39 
.39 
.40 
.40 

.270779 
.271479 
.272178 
.272876 
.273573 
.274269 

11  .69 
11.67 
11.65 
11.64 
11.62 
11.60 
11.58 

.729221 
.728521 
.727822 
.727124 
.726427 
.725731 

26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.267395 
.268065 
.268734 
.269402 
.270069 
.270735 
.271400 
.272064 
.272726 
.273388 

11.17 
11.15 
11.13 
11.12 
11.11 
11.08 
11.06 
11.05 
11.03 
11.01 

9.992430 

.992406 
.992382 
.992359 
.992335 
.992311 
.992287 
.992263 
.992239 
.992214 

.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 

9.274964 
.275658 
.276351 
.277043 
.277734 
.278424 
.279113 
.279801 
.280488 
.281174 

11.57 
11.55 
11.53 
11.51 
11.50 
11.48 
11.47 
11.45 
11.43 
11.41 

10.725036 
.724342 
.723649 
.722957 
.722266 
.721576 
.720887 
.720199 
.719512 
.718826 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
60 

9.274049 
.274708 
.275367 
.276025 
.276681 
.277337 
.277991 
.278644 
.279297 
.279948 
.280599 

10.99 
10.98 
10.96 
10.94 
10.92 
10.91 
10.89 
10.  8T 
10.86 
10.84 

9.992190 
.992166 
.992142 
.992118 
.992093 
.992069 
.992044 
.992020 
.991996 
.991971 
.991947 

.40 
.40 
.40 
.41 
.41 
.41 
.41 
.41 
.41 
.41 

9.281858 
.282542 
.283225 
.283907 
.284588 
.285268 
.285947 
.286624 
.287301 
.287977 
.288652 

11.40 
11.38 
11.36 
11.35 
11.33 
11.31 
11.30 
11.28 
11.26 
11.25 

10.718142 
.717458 
.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 
.712023 
.711348 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.  1". 

Sine. 

D.I  '. 

Cotang. 

D.  1   . 

Tang. 

M. 

100° 


79° 


TABLE  IT.      LOGAKITHMIC  SINES,  ETC.                  51 

11°                                                                                                                       168° 

M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 
6 
7 
8 

9.280599 
.281248 
.281897 
.282544 
.283190 
.283836 
.284480 
.285124 
.285766 

10.82 
10.81 
10.79 
10.77 
10.76 
10.74 
10.72 
10.71 

9.991947 
.991922 
.991897 
.991873 
.991848 
.991823 
.991799 
.991774 
.991749 

.41 
.41 
.41 
.41 
.41 
.41 
.41 
.42 
49 

9.288652 
.289326 
.289999 
.290671 
.291342 
.292013 
.292682 
.293350 
.294017 

11.23 
11.22 
11.20 
11.18 
11.17 
11.15 
11.14 
11.12 

10.711348 
.710674 
.710001 
.709329 
.708658 
.707987 
.707318 
.706650 
.705983 

60 
59 
68 
67 
66 
65 
64 
53 
62 

9 

.286408 

10.67 

.991724 

.42 

.294684 

11.09 

.705316 

61 

10 
11 
12 
13 
14 

9.287048 
.287688 
.288326 
.288964 
.289600 

10.66 
10.64 
10.63 
10.61 

9.991699 
.991674 
.991649 
.991624 
.991599 

.42 
.42 
.42 

.42 

9.295349 
.296013 
.296677 
.297339 
.298001 

11.07 
11.06 
11.04 
11.03 

10.704651 
.703987 
.703323 
.702661 
.701999 

60 
49 
48 
47 
46 

15 
16 
17 
18 
19 

.290236 
.290870 
.291504 
.292137 
.292768 

10.58 
10.56 
10.55 
10.53 
10.51 

.991574 
.991549 
.991524 
.991498 
.991473 

.42 
.42 
.42 
.42 
.42 

.298662 
.299322 
.299980 
.300638 
.301295 

11.00 
10.98 
10.97 
10.95 
10.93 

.701338 
.700678 
.700020 
.699362 
.698706 

45 
44 

43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.293399 
.294029 
.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.299034 

10.50 
10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.39 
10.37 
10.36 

9.991448 
.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 

.42 
.42 
.42 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

9.301951 
.302607 
.303261 
.303914 
.304567 
'  .305218 
.305869 
.306519 
.307168 
.307816 

10.92 
10.90 
10.89 
10.87 
10.86 
10.84 
10.83 
10.81 
10.80 
10.78 

10.698049 
.697393 
.696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 

35 
36 
37 
38 
39 

9.299655 
.300276 
.300895 
.301514 
.302132 
.302748 
.303364 
.303979 
.304593 
.305207 

10.34 
10.33 
10.31 
10.30 
10.28 
10.26 
10.25 
10.23 
10.22 
10.20 

9.991193 
.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.990986 
.990960 

.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

9.308463 
.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 
.314247 

10.77 
10.76 
10.74 
10.73 
10.71 
10.70 
10.68 
10.67 
10.65 
10.64 

10.691537 
.690891 
.690246 
.689601 
.688958 
.688315 
.687673 
.687032 
.686392 
.685753 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.305819 
.306430 
.307041 
.307650 
.308259 
.308867 
.309474 
.310080 
.310685 
.311289 

10.19 
10.17 
10.16 
10.14 
10.13 
10.12 
10.10 
10.09 
10.07 
10.06 

9.990934 
.990908 
.990882 
.990855 
.990829 
.990803 
.990777 
.990750 
.990724 
.990697 

.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 

9.314885 
.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
.319329 
.319961 
.320592 

10.62 
10.61 
10.60 
10.58 
10.57 
10.55 
10.54 
10.53 
10.51 
10.50 

10.685115 
.684477 
.683841 
.683205 
.682570 
.681936 
.681303 
.680671 
.680039 
.679408 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
51 
62 
63 
64 
65 
66 
67 
68 
69 
60 

9.311893 
.312495 
.313097 
.313698 
.314297 
.314897 
.315495 
.316092 
.316689 
.317284 
.317879 

10.04 
10.03 
10.01 
10.00 
9.98 
9.97 
9.96 
9.94 
9.93 
9.91 

9.990671 

.990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 
.990458 
.990431 
.990404 

.44 
.44 
.44 
.44 
.44 
.44 
.45 
.45 
.45 
.45 

9.321222 
.321851 
.322479 
.323106 
.323733 
.324358 
.324983 
.325607 
.326231 
.326853 
.327475 

10.48 
10.47 
10.46 
10.44 
10.43 
10.41 
10.40 
10.39 
10.37 
10.36 

10.678778 
.678149 
.677521 
.676894 
.676267 
.675642 
.675017 
.674393 
.673769 
.673147 
.672525 

10 
9 
8 

7 
6 
6 

4 
3 
2 
1 
0 

M. 

Cosine. 

D.  1". 

Sine. 

D.I'. 

Cotang. 

D.I". 

Tang. 

M. 

101°                                                                                                                 78* 

52 


TABLE  IT.      LOGABITHMIC  SINES,  ETC. 


M 

Sine. 

D.I" 

Cosine 

D.r 

Tang. 

D.I" 

Cotang 

M. 

{ 

9.317879 
.318473 
.319066 
.319658 
.320249 
.320840 

9.90 

9.88 
9.87 
9.86 
9.84 

9.990404 
.990378 
.990351 
.990324 
.990297 
.990270 

.4 
.4 

.45 
.45 
.45 

9-.  327475 
.32809 
.328715 
.329334 
.329953 
.330570 

10.35 
10.3 
10.3 
10.3 
10.29 

10.67252 
.67190 
.67128 
.67066 
.67004 
•  66943 

60 
69 

68 
57 
56 
65 

( 

.321430 

9  82 

.990243 

.45 

.331187 

10.28 

.66881 

54 

9 

.322019 
.322607 
.323194 

9.80 
9.79 
9.77 

.990215 
.990188 
.990161 

Ao 
.45 
.45 

.331803 
.332418 
.333033 

10.27 
10.25 
10.24 
10  23 

.66819 
.66758 
.66696 

63 
62 
61 

10 

1 

16 
17 

18 
19 

9.323780 
.324366 
.324950 
.325534 
.326117 
.326700 
.327281 
.327862 
.328442 
.329021 

9.76 
9.75 
9.73 
9.72 
9.70 
9.69 
9.68 
9.66 
9.65 
9.64 

9.990134 
.990107 
.990079 
.990052 
.990025 
.989997 
.989970 
.989942 
.989915 
.989887 

.45 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 

9.333646 
.334259 
.334871 
.335482 
.336093 
.336702 
.337311 
.337919 
.338527 
.339133 

10.21 
10.20 
10.19 
10.17 
10.16 
10.15 
10.14 
10.12 
10.11 
10  10 

10.66635 
.66574 
.66512 
.664518 
.663907 
.663298 
.662689 
.662081 
.661473 
.660867 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 

22 
23 
24 

25 
26 
27 
28 
29 

9.329599 
.330176 
.330753 
.331329 
.331903 
.332478 
.333051 
.333624 
.334195 
.334767 

9.62 
9.61 
9.60 
9.58 
9.57 
9.56 
9.54 
9.53 
9.52 
9.50 

9.989860 
.989832 
.989804 
.989777 
.989749 
.989721 
.989693 
.989665 
.989637 
.989610 

.46 
.46 
.46 
.46 
.47 
.47 
.47 
.47 
.47 
.47 

9.339739 
.340344 
.340948 
.341552 
.342155 
.342757 
.343358 
.343958 
.344558 
.345157 

10.08 
10.07 
10.06 
10.05 
10.03 
10.02 
10.01 
10.00 
9.98 
9  97 

10.660261 
.659656 
.659052 
.658448 
.657845 
.657243 
.656642 
.656042 
.655442 
.654843 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.335337 
.335900 
.336475 
.337043 
.337610 
.338176 
.338742 

9.49 
9.48 
9.46 
9.45 
9.44 
9.43 
9.41 

9.989582 
.989553 
.989525 
.989497 
.989469 
.989441 
.989413 

.47 
.47 
.47 
.47 
.47 
.47 
47 

9.345755 
.346353 
.346949 
.347545 
.348141 
.348735 
.349329 

9.96 
9.95 
9.93 
9.92 
9.91 
9.90 

10.654245 
.653647 
.653051 
.652455 
.651859 
.651265 
.650671 

30 
29 

28 
27 
26 
25 
24 

38 
39 

.339871 
.340434 

9.40 
9.39 
9.37 

.989385 
.989356 
.989328 

.47 
.47 

.47 

.349922 
.350514 
.351106 

9.87 
9.86 

9oe 

.650078 
.649486 
.648894 

23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.340996 
.341558 
.342119 
.342679 
.343239 
.343797 
.344355 
.344912 
.345469 
.346024 

9.36 
9.35 
9.34 
9.32 
9.31 
9.30 
9.29 
9.27 
9.26 
9.25 

9.989300 
.989271 
.989243 
.989214 
.989186 
.989157 
.989128 
.989100 
.989071 
.989042 

.47 

.47 
.47 

.47 
.47 

.47 

.48 
.48 
.48 
48 

9.351697 
.352287 
.352876 
.353465 
.354053 
.354640 
.355227 
.355813 
.356398 
.356982 

9.84 
9.82 
9.81 
9.80 
9  79 
9.78 
9.76 
9.75 
9.74 

0.648303 
.647713 
.647124 
.646535 
.645947 
.645360 
.644773 
.644187 
.643602 
.643018 

20 

19 
18 
17 
16 
15 
14 
13 
2 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.346579 
.347134 
.347687 
.348240 
.348792 
.349343 
.349893 
.350443 
.350992 
.351540 
.352088 

9.24 
9.22 
9.21 
9.20 
9.19 
9.17 
9.16 
9.15 
9.14 
9.13 

9.989014 

.988985 
.988956 
.988927 
.988898 
.988869 
.988840 
.988811 
.988782 
.988753 
.988724 

.48 
.48 
.48 
.48 
.48 
.48 
.48 
.49 
.49 
.49 

9.357566 
.358149 
.358731 
.359313 
.359893 
.360474 
.361053 
.361632 
.362210 
.362787 
.363364 

9.72 
9.70 
9.69 
9.68 
9.67 
9.66 
9.65 
9.63 
9.62 
9.61 

0.642434 
.641851 
.641269 
.640687 
.640107 
.639520 
.638947 
.638368 
.637790 
.637213 
.636636 

0 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine,  j 

D.I". 

Sine. 

.1'. 

Cotang. 

D.I  . 

Tang. 

TABLE  IV.      LOGARITHMIC  SINES,  ETC. 


18° 


166* 


M.       Sine.       D.l   .     Cosine.     D.r       Tang.      D.I".     Cotang.    M. 


9.352088 
.352635 
.353181 
.353726 
.354271 
.354815 
.355358 
.355901 
.356443 


9.357524 
.358064 
.358603 


.359678 
.360215 
.360752 
.361287 
.361822 


.363422 
.363954 
.364485 
.365016 
.365546 


.367131 
.367659 

9.368185 
.368711 


.369761 
.370285 
.370808 
.371330 
.371852 
.372373 


9.373414 
.373933 
.374452 
.374970 
.375487 
.376003 
.376519 
.377035 
.377549 
.378063 

9.378577 
.379089 
.379601 
.380113 


.381134 
.381643 
.382152 


.383168 
.383675 


9.11 
9.10 


9.07 
9.05 
9.04 
9.03 
9.02 
9.01 

8.99 
8.98 
8.97 
8.96 
8.95 
8.94 
8.92 
8.91 
8.90 


8.87 
8.86 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

8.76 
8.75 
8.74 
8.73 
8.72 
8.71 
8.70 


8.65 
8.64 
8.63 
8.62 
8.61 


8.58 
8.57 
8.56 

8.55 
8.53 
8.52 
8.51 
8.50 
8.49 
8.48 
8.47 
8.46 
8.45 


9.988724 


.988578 
.988548 
.988519 


.988371 


.988193 
.988163 

9.988133 


.988073 
.988043 
.988013 


.987953 
.987922 
.987892 
.987862 

9.987832 
.987801 
.987771 
.987740 
.987710 
.987679 
.987649 
.987618 
.987588 
.987557 

9.987526 


.987465 
.987434 
.987403 
.987372 
.987341 
.987310 
.987279 
.987248 

.987217 
.987186 
.987155 
.987124 
.987092 
.987061 
.987030 


.49 
.49 
.49 
.49 
.49 
.49 
.49 
.49 
.49 

.49 
.49 
.49 
.49 
.50 
.50 
.50 
.50 
.50 
.50 

.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 

.51 

.51 
.51 
.51 
.51 
.51 
.51 
.51 
.51 
.51 

.61 
.51 
.51 
.51 
.52 
.52 
.52 
.52 
.52 
.52 

.52 


.52 
.52 


.52 


.52 
.52 


.364515 
.365090 


.986237 


.367382 
.367953 


9.369094 


.370232 
.370799 
.371367 
.371933 
.372499 
.373064 
.373629 
.374193 

9.374756 
.375319 


.376442 
.377003 
.377563 
.378122 


.379797 


.380910 
.381466 

.382020 
.382575 
.383129 


.3*4234 
.384786 


.389178 


.390270 
.390815 


.391903 

.392447 


.394073 
.394614 
.395154 


.396771 


9.60 
9.59 
9.58 
9.57 
9.55 
9.54 
9.53 
9.52 
9.51 
9.50 

9.49 
9.48 
9.47 
9.45 
9.44 
9.43 
9.42 
9.41 
9.40 
9.39 

9.38 
9.37 
9.36 
9.35 
9.33 
9.32 
9.31 
9.30 
9.29 
9.28 

9.27 
9.26 
9.25 
9.24 
9.23 
9.22 
9.21 
9.20 
9.19 
9.18 

9.17 
9.16 
9.15 
9.14 
9.12 
9.11 
9.10 


9.07 

9.06 
9.05 
9.04 
9.03 
9.02 
9.01 
9.00 


8.97 


10.636636 
.636060 
.635485 
.634910 
.634336 
.633763 
.633190 
.632618 
.632047 
.631476 

10.630906 
.630337 
.629768 
.629201 
.628633 
.628067 
.627501 
.626936 
.626371 
.625807 

10.625244 
.624681 
.624119 
.623558 
.622997 
.622437 
.621878 
.621319 
.620761 
.620203 

10.619646 
.619090 
.618534 
.617980 
.617425 
.616871 
.616318 
.615766 
.615214 
.614663 

10.614112 
.613562 
.613013 
.612464 
.611916 
.611369 
.610822 
.610276 
.609730 


.607553 
.607011 


.604846 
.604306 
.603767 


M.  1  Cosine.     D.l".        Sine.     D.l".    Cotang.    D.l".       Tang.     M. 
°  29  W 


54: 


TABLE   IV.      LOGARITHMIC   SINES,  ETC. 


165" 


M. 

Sine. 

D.l". 

Cosine. 

D.I" 

Tang. 

D.l". 

Cotang. 

M. 

0 
1 

2 
3 
4 
6 
6 
7 
8 

9.383675 

.384182 
.384687 
.385192 
.385697 
.386201 
.386704 
.387207 
.387709 

8.44 
8.43 

8.42 
8.41 
8.40 
8.39 
8.38 
8.37 

9.986904 
.986873 
.986841 
.986809 
.986778 
.986746 
.986714 
.986683 
.986651 

.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 

9.396771 
.397309 
.397846 
.398383 
.398919 
.399455 
.399990 
.400524 
.401058 

8.96 
8.96 
8.95 
8.94 
8.93 
8.92 
8.91 
8.90 

10.603229 
.602691 
.602154 
.601617 
.601081 
.600545 
.600010 
.599476 
598942 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 

.388210 

8.35 

.986619 

.53 

.401591 

8.89 
8  88 

.598409 

51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.388711 
.389211 
.389711 
.390210 
.390708 
.391206 
.391703 
.392199 
.392695 
.393191 

8.34 
8.33 
8.32 
8.31 
8.30 
8.28 
8.27 
8.26 
8.25 
8.24 

9.986587 
.986555 
.986523 
.986491 
.986459 
.986427 
.986395 
.986363 
.986331 
.986299 

.53 
.53 
.53 
.53 
.53 
.53 
.53 
.54 
.54 
.64 

9.402124 
.402656 
.403187 
.403718 
.404249 
.404778 
.40530§ 
.405836 
.406364 
.406892 

8.87 
8.86 
8.85 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

10.597876 
.597344 
.596813 
.596282 
.595751 
.595222 
.594692 
.594164 
.593636 
.593108 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 

9.393685 
.394179 
.394673 
.395166 
.395658 
.396150 
.396641' 
.397132 

8.23 

8.22 
8.21 
8.20 
8.19 
8.18 
8.17 

9.986266 
.986234 
.986202 
.986169 
.986137 
.986101 
.986072 
.986039 

.54 
.54 
.54 
.54 
.54 
.54 
.54 

9.407419 

.407945 
.408471 
.408997 
.409521 
.410045 
.410569 
.411092 

8.77 
8.76 
8.75 
8.74 
8.74 
8.73 
8.72 

10.592581 
.592055 
.591529 
.591003 
.590479 
.589955 
.589431 
588908 

40 

39 
38 
37 
36 
35 
34 
33 

28 
29 

.397621 
.398111 

8.16 
8.15 

.986007 
.985974 

.54 

.54 
.54 

.411615 
.412137 

8.71 
8.70 
8  69 

.588385 
.587863 

32 
31 

30 

9.398600 

8.14 

9.985942 

54 

9.412658 

8  68 

10.587342 

30 

81 
82 
83 
84 
85 
86 
37 
38 
89 

.399088 
.399575 
.400062 
.400549 
.401035 
.401520 
.402005 
.402489 
,402972 

8.13 
8.12 
8.11 
8.10 
8.09 
8.08 
8.07 
8.06 
8.05 

.985909 
.985876 
.985843 
.985811 
.985778 
.985745 
.985712 
.985679 
.985646 

.55 
.55 
.55 
.55 
.55 
.55 
.55 
.55 
.55 

.413179 
.413699 
.414219 
.414738 
.415257 
.415775 
.416293 
.416810 
.417326 

8.67 
8.66 
8.65 
8.65 
8.64 
8.63 
8.62 
8.61 
8.60 

.586821 
.586301 
.585781 
.585262 
.584743 
.584225 
.583707 
.583190 
.582674 

29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.403455 
.403938 
.404420 
.404901 
.405382 
.405862 
.406341 
.406820 
.407299 
.407777 

8.04 
8.03 
8.02 
8.01 
8.00 
7.99 
7.98 
7.97 
7.96 
7.95 

9.985613 
.985580 
.985547 
.985514 
.985480 
.985447 
.985414 
.985380 
.985347 
.985314 

.55 
.55 
.55 
.55 
.55 
.55 
.56 
.56 
.56 
.56 

9.417842 

.418358 
.418873 
.419387 
.419901 
.420415 
.420927 
.421440 
.421952 
.422463 

8.59 
8.58 
8.57 
8.56 
8.56 
8.55 
8.54 
8.53 
8.52 
8  51 

10.582158 
.581642 
.581127 
.580613 
.580099 
.579585 
.579073 
.578560 
.578048 
.577537 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 

61 
62 
63 
64 
65 
66 
67 
68 
69 
60 

9.408254 
.408731 
.409207 
.409682 
.410157 
.410632 
.411106 
.411579 
.412052 
.412524 
.412996 

7.94 

7.94 
7.93 
7.92 
7.91 
7.90 
7.89 
7.88 
7.87 
7.86 

9.985280 
.985247 
.985213 
.985180 
.985146 
.985113 
.985079 
.985045 
.985011 
.984978 
.984944 

.56 
.56 
.56 
.56 
.56 
.56 
.56 
.56 
.56 
.56 

9.422974 
.423484 
.423993 
.424503 
.425011 
.425519 
.426027 
.426534 
.427041 
.427547 
.428052 

8.50 
8.49 
8.49 
8.48 
8.47 
8.46 
8.45 
8.44 
8.43 
8.43 

10.577026 
.576516 
.576007 
.575497 
.574989 
.574481 
.573973 
.573466 
.572959 
.572453 
.571948 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine, 

D.I". 

Sine. 

D.I". 

Cotang. 

D.l  ". 

Tang. 

M 

104'                                              7K. 

TABLE   IV.      LOGARITHMIC  SINES,   ETC.  55 

164° 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l  . 

Cotang. 

M. 

0 
1 

9.412996 
.413467 

7.85 

7  R4 

9.984944 
.984910 

.57 

9.428052 

.428558 

8.42 

10.571948 
.571442 

60 
59 

2 
3 

.413938 
.414408 

7i84 

.984876 
.984842 

'.57 

KTT 

.429062 
.429566 

S.40 

.570938 
.570434 

58 
'57 

4 

.414878 

7.83 

700 

.984808 

.57 

CT 

.430070 

2  » 

.569930 

56 

•  6 
6 

.415347 
.415815 

.  O-rfJ 

7.81 

7OA 

.984774 
.984740 

•  Oi 

.57 

KT 

.430573 
.431075 

8i38 

807 

.569427 
.568925 

55 
54 

7 

.416283 

.oU 

.984706 

•  O« 

KfJ 

.431577 

•  Of 

8o/» 

.568423 

53 

8 

.416751 

7*  7ft 

.984672 

•  Ol 

*7 

.432079 

.OO 

8QK 

.567921 

62 

9 

.417217 

4  .  4O 

7.77 

.984637 

•  Ol 

.57 

.432580 

.  oO 

8.34 

.567420 

51 

10 
11 
12 

9.417684 
.418150 
.418615 

7.76 
7.75 

77K 

9.984603 
.984569 
.984535 

.57 
.57 

eff 

9.433080 
.433580 
.434080 

8.33 
8.33 

10.566920 
.566420 
.565920 

50 
49 

48 

13 
14 
15 

.419079 
.419544 

.420007 

.  45 

7.74 
7.73 

.984500 
.984466 
.984432 

.Ol 

.67 

.57 

CO 

.434579 
.435078 
.435576 

sisi 

8.30 

,565421 
.564922 
.564424 

47 
46 
45 

16 

.420470 

7.72 

771 

.984397 

.5o 

KQ 

.436073 

«  98 

.563927 

44 

17 

.420933 

.  ll 

.984363 

•  Do 

•ro 

.436570 

a  oa 

.563430 

43 

18 

.421395 

7.70 

7  fiQ 

.984328 

.5o 

CO 

.437067 

8*07 

.562933 

42 

19 

.421857 

»  .oy 
7.68 

.984294 

.Do 

.58 

.437563 

.£i 

8.23 

.562137 

41 

20 

9.422318 

9.984259 

KQ 

9.438059 

10.561941 

40 

21 
22 

.422778 
.423238 

7.67 
7.67 

Tea 

.984224 
.984190 

.Do 
.58 

.438554 
.439048 

8.  '24 

8O/4 

.561446 
.560952 

39 

38 

23 

.423697 

•  DO 
Tfift 

.984155 

CO 

.439543 

»£& 

0  00 

.560457 

37 

24 
25 

.424156 
.424615 

•  DO 

7.64 

7/jO 

.984120 
.984085 

.Do 

.58 

KQ 

.440036 
.440529 

o.  _•> 

8.22 
801 

.559964 
.559471 

36 
35 

26 

.425073 

.DO 

7s«o 

.984050 

.on 

.441022 

.£l 

8Ort 

.558978 

34 

27 

.425530 

.62 

7(M 

.984015 

.58 

CO 

.441514 

.20 

8OA 

.558486 

33 

28 

.425987 

•  bl 

7*1 

.983981 

.Do 

.442006 

.  A) 

8-m 

.557994 

32 

29 

.426443 

•  61 
7.60 

.983946 

.58 
.58 

.442497 

.19 
8.18 

.557503 

31 

30 

9.426899 

7KCL 

9.983911 

9.442988 

817 

10.557012 

30 

31 

.427354 

.5y 

7CQ 

.983875 

.58 

CO 

.443479 

.17 
8-1  jt» 

.556521 

29 

32 

.427809 

,5o 

7KT 

.983840 

.Do 

.443968 

.  lo 

81  ^ 

.556032 

28 

33 
34 

.428263 
.428717 

.57 
7.56 

7Cf» 

.983805 
.983770 

.59 
.59 

KO 

.444458 
.444947 

.lo 

8.15 
81  j 

.555542 
.555053 

27 
26 

35 
36 
37 

.429170 
.429623 
.430075 

.55 
7.54 
7.53 

.983735 
.983700 
.983664 

.59 
.59 
.59 

.445435 
.445923 
.446411 

.14 

8.13 
8.13 

C  1  •> 

.554565 
.554077 
.553589 

25 
24 
23 

38 

.430527 

7.52 

7  CO 

.983629 

.59 

.446898 

.553102 

22 

39 

.430978 

.52 
7.51 

.983594 

.59 
.59 

.447384 

8.11 
8.10 

.552616 

21 

40 

9.431429 

7  Kfl 

9.983558 

9.447870 

800 

10.552130 

20 

41 

.431879 

4  .DU 

.983523 

.59 

.448356 

.uy 

8AO 

.551644 

19 

42 

.432329 

7.49 

.983487 

.59 

.448841 

.uy 

O  AQ. 

.551159 

18 

43 

.432778 

7.49 

7   JO 

.983452 

-.59 

•449326 

o.uo 

o  /yr 

.550674 

17 

44 

45 

.433226 
.433675 

,4o 
7.47 

7  -ir 

.983416 
.983381 

!59 

.449810 
.450294 

O.U4 

8.06 

.550190 
.549706 

16 
15 

46 
47 

.434122 
.434569 

7^45 

7   MM 

.983345 
.983309 

.59 
.59 

.450777 
.451260 

8.06 
8.05 

.549223 
.548740 

14 
13 

48 
49 

.435016 
.435462 

.44 
7.44 
7.43 

.983273 
.983238 

.69 
.60 
.60 

.451743 
.452225 

8.04 
8.03 
8.03 

.548257 
.547775 

12 
11 

60 

9.435908 

7  An 

9.983202 

9,452706 

10.547294 

10 

61 

.436353 

.42 

741 

.983166 

.60 

.453187 

8.02- 

8A1 

.546813 

9 

62 
63 

.436798 
.437242 

.41 

7.40 

7>IA 

.983130 
.983094 

.60 
.60 

.453668 
.454148 

.01 
8.00 

.546332 
.545852 

8 

7 

64 

.437686 

.40 

700 

.983058 

.60 

.454628 

8.00 

.545372 

6 

65 

.438129 

.0*7 

TOO 

.983022 

.60 

.455107 

7.99 

.544893 

*5 

56 

.438572 

.00 

.982986 

.60 

.455586 

7.98 

.544414 

4 

57 

.439014 

7.37 

70£ 

.982950 

.60 

.456064 

7.97 

.543936 

3 

68 

.439456 

•  OO 

.982914 

.60 

.456542 

7.97 

.543458 

2 

59 

.439897 

7.36 

7  OK 

.982878 

.60 

.457019 

7nK 

.542981 

1 

60 

.440338 

.OO 

.982842 

.60 

.457496 

.95 

.542504 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l  . 

Cotang. 

D.l". 

Tang. 

M. 

1O5°                                             740 

56 


TABLE  IV.      LOGARITHMIC  SINES,  ETC. 


163° 


M. 

Sine. 

D.I".  |  Cosine. 

D  1" 

Tang. 

D.1". 

Cotang. 

M. 

0 

9.440338 

9.982842 

9.457496 

7  94 

10.542504 

60 

1 

2 
3 
4 
5 
6 
7 

.440778 
.441218 
.441658 
.442096 
.442535 
.442973 
.443410 

7^33 
7.32 
7.31 
7.31 
7.30 
7.29 

7OQ 

.982805 
.982769 
.982733 
.982696 
.982660 
.982624 
.982587 

!co 

.61 
.61 
.61 

.61 
.61 
d 

.457973 
.458449 
.458925 
.459400 
.459875 
.460349 
.460823 

7.  '94 
7.93 
7.92 
7.91 
7.91 
7.90 

.542027 
.541551 
.541075 
.540600 
.540125 
.539651 
.539177 

59 

58 
57 
56 
55 
54 
53 

8 

.443847 

.Jo 

.982551 

•  Ol 

.461297 

7QQ 

.538703 

52 

9 

.444284 

7.27 
7.27 

.982514 

.61 
.61 

.461770 

.00 

7.88 

.538230 

51 

10 
11 

9.444720 
.445155 

7.26 

9.982477 
.982441 

.61 

9.462242 
.462714 

7.87 

7QA 

10.537758 
.537286 

eo 

49 

12 

.445590 

7.25 

.982404 

M 

.463186 

.  OO 
7QA 

.536814 

48 

13 
14 
15 
16 

.446025 
.446459 
.446893 
.447326 

7.  24 
7.24 
7.23 
7.22 
7  21 

.982367 
.982331 
.982294 

.982257 

•  Ol 

.61 
.61 
.61 

A1 

.463658 
.464129 
.464599 
.465069 

.OO 

7.85 
7.84 
7.83 

7QO 

.536342 
.535871 
.535401 
.534931 

47 
46 
45 
44 

17 
18 
19 

.447759 
.448191 
.448623 

7l20 
7.20 
7.19 

.982220 
.982183 
.982146 

.01 

.62 
.62 
.62 

.465539 
.466008 
.466476 

.OO 

7.82 
7.81 
7.81 

.534461 
.533992 
.533524 

43 

42 
41 

20 

9.449054 

71  Q 

9.982109 

9.466945 

10.533055 

40 

21 

.449485 

.lo 

717 

.982072 

fi9 

.467413 

77Q 

.532587 

39 

22 
23 
24 

.449915 
.450345 
.450775 

.  14 

7.17 
7.16 

71  £. 

.982035 
.981998 
.981961 

'.62 
.62 

.467880 
.468347 
.468814 

.  i  J 

7.78 
7.78 

777 

.532120 
.531653 
.531186 

38 
37 
36 

25 
26 

.451204 
.451632 

.  10 

7.14 

71  Q 

.981924 
.981886 

^62 

AO 

.469280 
.469746 

.  4  i 

7.76 

77A 

.530720 
.530254 

35 
34 

27 
28 

.452060 

.452488 

.  lo 

7.13 

7  19 

.981849 
.981812 

ftvQI 

.62 

AO 

.470211 
.470676 

.  (0 

7.75 
7  74 

.529789 
.529324 

33 
32 

29 

.452915 

4  .  1  w 

7.11 

.981774 

•yQI 

.62 

.471141 

4  .  I  'i 

7.74 

.528859 

31 

30 
31 

9.453342 
.453768 

7.10 

71A 

9.981737 
.981700 

.62 

9.471605 
.472068 

7.73 

10.528395 
.527932 

20 

29 

32 
33 

.454194 
.454619 

.  l\) 

7.09 

.981662 
.981625 

!63 

.472532 
.472995 

Tin 

.527468 
.527005 

28 

27 

34 

.455044 

7.08 

.981587 

.63 

AQ 

.473457 

7.71 

77A 

.526543 

26 

35 

.455469 

7*07 

.981549 

.OO 

.473919 

.  lU 

7AQ 

.526081 

25 

36 

.455893 

1  .1/4 
7A/» 

.981512 

•™ 

.474381 

.  oy 

7AQ 

.525619 

24 

37 

.456316 

.Ut> 
7f\K 

.981474 

/.Q 

.474842 

.oy 

7/»Q 

.525158 

23 

38 

.456739 

.(JO 

.981436 

.DO 

.475303 

.Oo 

7A7 

.524697 

22 

39 

.457162 

7!04 

.981399 

'.63 

.475763 

.  Ol 

7.67 

.524237 

21 

40 
41 
42 
43 

9.457584 
.458006 
.458427 
•_  .458848 

7.03 

7.02 
7.01 

9.981361 
.981323 

.981285 
.981247 

.63 
.63 
.63 

AQ 

9.476223 
.476683 
.477142 
.477601 

7.66 
7.65 
7.65 

7AA 

10.523777 
.523317 
.522858 
.522399 

20 
19 
18 
17 

41 

.459268 

7AA 

.981209 

•  Oo 

.478059 

.O'l 

7AO. 

.521941 

10 

4> 

46 

.459688 
.460108 

.UU 

6.99 

.981171 
.981133 

'.63 

/»o 

.478517 
.478975 

.Oo 
7.63 

.521483 
.521025 

15 
14 

47 

.460527 

6  no 

.981095 

•  Oo 

.479432 

7/21 

.520568 

13 

43 

.460946 

.y<5 

6G7 

.981057 

AA 

.479889 

.01 

7A1 

.520111 

12 

49 

.461364 

.y  i 
6.96 

.981019 

.O'l 

.64 

.480345 

.  01 

7.60 

.519655 

11 

EO 

9.461782 

9.980981 

AA 

9.480801 

7 

10.519199 

10 

51 

.462199 

«   K. 

.980942 

»wB 

.481257 

7.59 

.518743 

9 

52 

.462616 

C'od 

.980904 

AA. 

.481712 

7EQ 

.518288 

8 

53 

.463032 

.U4 

.980866 

.0% 

.482167 

.Oo 

7K7 

.517833 

7 

5i 
55  « 

.463448 
.463864 

e!93 

.980827 
.980789 

'.64 

Al 

.482621 
.483075 

.Ol 

7.57 

7KA 

.517379 
.516925 

6 
5 

56 
57 

.464279 
.464694 

6.  '91 

6QA 

.980750 
.980712 

.01 

.64 

AA 

.483529 
.483982 

.00 

7.55 

7KK 

.516471 
.516018 

4 
3 

58 
59 

.465108 
.465522 

.  yu 
6.90 

.980073 
.980635 

.0* 

.64 
fid 

.484435 

.484887 

.00 

7.54 

7  CO 

.515565 
.515113 

2 
1 

60 

.465935 

' 

.980596 

.04 

.485339 

.Oo 

.  51466  I 

0 

M. 

Cosine. 

D.I  . 

Sine. 

D.I". 

Cotang. 

D.I  '. 

Tang. 

M. 

106°                                             73° 

I7C 


TABLE  IV.      LOGARITHMIC  SINES,   ETC.  67 

162° 


M. 

Sine. 

D.l". 

Cosine.  |D.l". 

Tang. 

D.I". 

Cotang. 

M. 

0 

1 
2 
3 

9.465935 
.466348 
.466761 
.467173 

6.88 
6.88 
6.87 

6  on 

9.980596 
.980558 
.980519 
.980480 

.64 
.64 
.65 

OK 

9.485339 

.48.3791 
.480242 
.486693 

7.53 
7.52 
7.51 
T  Pil 

0.514661 
.514209 
.513758 
.513307 

60 
59 
68 
57 

4 
5 
6 

7 
8 
9 

.467585 
.467996 
.468407 
.468817 
.469227 
.469637 

.  oO 

6.85 
6.85 
6.84 
6.83 
6.83 
6.82 

.980442 
.980403 
.980364 
.980325 
.980286 
.980247 

•  OD 

.65 
.65 
.65 
.65 
.65 
.65 

.487143 
.487593 
.488043 
.488492 
.488941 
.489390 

t  .01 
7.50 
7.50 
7.49 
7.48 
7.48 
7.47 

.512857 
.612407 

.511957 
.511508 
.511059 
.510610 

56 
55 
54 
53 
52 
51 

10 
11 
12 
13 

9,470046 
.470455 
.470863 
.471271 

6.81 
6.81 
6.80 

9.980208 
.980169 
.980130 
.980091 

.65 

.65 
.65 

9.489838 

.490286 
.490733 
.491180 

7.46 
7.46 
7.45 

0.510162 
.509714 
.509267 
.508820 

50 
49 
48 
47 

14 
15 
16 

.471679 
.472086 
.472492 

6.79 
6.78 
6.78 

.980052 
.980012 
.979973 

.65 
.65 
.65 

.491627 
.492073 
.492519 

7.44 
7.44 
7.43 

.508373 
.507927 
.507481 

46 
45 
44 

17 

.472898 

6.77 

6  'ret 

.979934 

.65 

act 

.492965 

7.43 
7j«) 

.507035 

43 

18 

.473304 

.7o 

.979895 

.DO 

.493410 

.d 

.506590 

42 

19 

.473710 

6.76 
6.75 

.979855 

.66 
.66 

.493854 

7.41 
7.41 

•506146 

41 

20 

9.474115 

6TJ 

9.979816 

9.494299 

74  A 

0.505701 

40 

21 
22 
23 

.474519 
.474923 

.475327 

.7* 
6.74 
6.73 

.979776 
.979737 
.979697 

.66 
.66 
.66 

.494743 
.495186 
.495630 

.4(1 
7.39 
7.39 

.505257 
.504814 
.504370 

39 
38 
37 

24 
25 

.475730 
.476133 

6  72 
6.72 

.979658 
.979618 

.66 
.66 

.496073 
.496515 

7.38 
7.38 

.503927 

36 
35 

26 

.476536 

6.71 

.979579 

.66 

.496957 

7.37 

!  50*H3 

34 

27 

.476938 

6.70 

.979539 

.66 

.497399 

7.36 

7o/-» 

.502601 

33 

28 

.477340 

6.69 

.979499 

.66 

.497841 

.00 

.502159 

32 

29 

.477741 

6.69 
6.68 

.979459 

.66 
.66 

.498282 

7.35 
7.34 

.501718 

31 

30 

9.478142 

9.979420 

9^498722 

10.501278 

30 

31 

.478542 

6.67 

.979380 

.66 

.499163 

7.34 

.500837 

29 

32 

.478942 

6.67 

.979340 

.66 

.499603 

7.33 

.500397 

28 

33 

.479342 

6.66 

.979300 

.67 

.500042 

7.33 

.499958 

27 

34 
35 
36 
37 

.479741 
.480140 
.480539 
.480937 

6.65 
6.65 
6.64 
6.63 

.979260 
.979220 
.979180 
.979140 

.67 
.67 
.67 
.67 

r*7 

.500481 
.500920 
.501359 
.501797 

7.32 
7.31 
7.31 
7.30 

.499519 
.499080 
.498641 
.498203 

26 
25 
24 
23 

38 

.481334 

6.63 

.979100 

.o7 

.502235 

7.30 

.497765 

22 

39 

.481731 

6.62 
6.61 

.979059 

.67 

.67 

.502672 

7.29 
7.28 

.497328 

21 

40 
41 
42 
43 
44 
45 
46 
47 

9.482128 
.482525 
.482921 
.483316 
.483712 
.484107 
.484501 
.484895 

6.61 
6.60 
6.59 
6.59 
6.58 
6.57 
6.57 

9.979019 
.978979 
.978939 

.978898 
.978sr,8 
.978817 
.978777 
.978736 

.67 
.67 
.67 
.67 
.67 
.67 
.67 

9.503109 

.503546 
.503982 
.504418 
.504854 
.505289 
.505724 
.506159 

7.28 
7.27 
7.27 
7.26 
7.25 
7.25 
7.24 

7O4 

10.496891 
.496454 
.496018 
.49558-2 
.495146 
.494711 
.494276 
.493841 

20 
19 
18 
17 
16 
15 
14 
13 

48 

.485289 

6.56 

.978696 

.68 

.506593 

.24 

.493407 

12 

49 

.485682 

6.55 
6.55 

.978655 

.68 
.68 

.507027 

7.23 
7.2S 

.492973 

11 

50 

9.486075 

6"  1 

9.978615 

f\Q 

9.507460 

TOO 

10.492540 

10 

51 
52 

.486467 
.486860 

-O4 

6.54 

.978574 
.978533 

•  Do 

.68 

.r> 

.507893 
.508326 

•XI 
7.21 

70-f 

.492107 
.491674 

9 

8 

53 

.487251 

6.53 

.978493 

.00 

.508759 

.zl 

.491241 

7 

54 

55 

.487643 
.488034 

6.52 
6.52 

.978452 
.978411 

.68 
.68 

.509191 
.509622 

7.20 
7.20 

.490809 
.490378 

6 
5 

56 
57 
58 

.488424 
.488814 
.489204 

6.51 
3.50 

6.50 

.978370 
.978329 
.978288 

.68 
.68 
.68 

.510054 
.510485 
.510916 

7.  19 
7.18 
7.18 

.489946 
.489515 
.489084 

4 
3 
2 

59 

.489593 

6.49 

3  AO 

.978247 

.68 

(•0 

.511346 

7.17 
717 

.488654 

1 

60 

.489982 

.978206 

.00 

.511776 

.14 

.488224 

0 

M. 

Cosine 

rD.i". 

Sine. 

r».v 

fotanp. 

D1  '. 

Tang. 

M. 

107C 


72C 


58     TABLE  IV.   LOGARITHMIC  SINES,  ETC. 

18D                                             161° 

M. 

Sine. 

D.l  . 

Cosine. 

D.I". 

Tang. 

D.I  ". 

Cotang,  M. 

0 

1 

9  .489982 
.490371 

6.48 
fi  47 

9.978206 

.978165 

.68 

9.511776 

.512206 

7.16 

7%m 

10.488224 

.487794 

60 

59 

2 

,490759 

.978124 

ro 

.512635 

.  lb 

.487365 

58 

3 

.491147 

6*4fi 

.978083 

f\Q 

.513064 

7.15 
71  1 

.486936 

67 

4 

.491535 

•  40 

.978042 

•  U.7 

.513493 

.14 

.486507 

56 

6 

.491922 

6*45 

.978001 

ViQ 

.613921 

7.14 

71  o 

.486079 

55 

6 

.492308 

6*44. 

.977959 

.  uy 

ftQ 

.514349 

.  lo 
71  o. 

.485651 

54 

7 

.492695 

•  4-» 

.977918 

•HP 

.514777 

.  lo 

.485223 

53 

8 

.493081 

6  43 

.977877 

ro 

.515204 

7.  12 

.484796 

52 

9 

.493466 

6.42 

.977835 

.'69 

.515631 

7^11 

.484369 

51 

10 
11 
12 

9.493851 
.494236 
.494621 

6.41 
6.41 

9.977794 
.977752 
.977711 

.69 
.69 

9.516057 
.516484 
.516910 

7.10 
7.10 

10.483943 
.483516 
.483090 

50 
49 
48 

13 

.495005 

5*15 

.977669 

.69 

.517335 

7.09 

.482665 

47 

14 

.495388 

?*  2 

.977628 

CO 

.517761 

7.09 

.482239 

46 

15 

.495772 

6*38 

.977586 

*2 

.518185 

7.08 

.481815 

45 

16 

.496154 

6OQ 

.977544 

•*jj 

.518610 

7.08 

.481390 

44 

17 
18 

.496537 
.496919 

.OO 

6.37 

60.fi 

.977503 
.977461 

!70 

.519034 
.519458 

7.07 
7.07 

.480966 
.480542 

43 

42 

19 

.497301 

.00 

6.36 

.977419 

!70 

.519882 

7.06 
7.05 

.480118 

41 

20 

9.497682 

6  OK 

9.977377 

9.520305 

10.479695 

40 

21 
22 
23 
24 
25 
26 

.498064 
.498444 
.498825 
.499204 
.499584 
.499963 

.oD 

6.34 
6.34 
6.33 
6.33 
6.32 

6Q1 

.977335 
.977293 
.977251 
.977209 
.977167 
.977125 

.70 
.70 
.70 
.70 
.70 

.520728 
.521151 
.521573 
.521995 
.522417 
.522838 

7.05 
7.01 
7.04 
7.03 
7.03 
7.02 

.479272 
.478849 
.478427 
.478005 
.477583 
.477162 

39 
38 
37 
36 
35 
34 

27 
28 
29 

.500342 
.500721 
.501099 

ol 

6.31 
6.30 
6.30 

.977083 
.977041 
.976999 

.70 
.70 
.70 
.70 

.523259 
.523680 
.524100 

7.02 
7.01 
7.01 
7.00 

.476741 
.476320 
.475900 

33 
32 
31 

30 
31 

9.501476 
.501854 

6.29 

9.976957 
.976914 

.70 

"71 

9.524520 
.524939 

6.99 

10.475480 
.475061 

30 

29 

32 

.502231 

«  oa 

.976872 

•  4  1 

.525359 

6.99 

.474641 

28 

33 

.502607 

6  '07 

.976830 

•  71 

.525778 

6.98 

.474222 

27 

34 
35 
36 

.502984 
.503360 
.503735 

.XI 

6.27 
6.26 

.976787 
.976745 
.976702 

•  71 
.71 
.71 

*71 

.526197 
.526615 
.527033 

6.98 
6.97 
6.97 

.473803 
.473385 
.472967 

26 
25 
24 

37 
38 
39 

.504110 
.504485 
.504860 

6^25 
6.24 
6.24 

.976660 
.976617 
.976574 

•  ll 

.71 

.71 
.71 

.527451 

.527868 
.528285 

6.96 
6.96 
6.95 
6.95 

.472549 
.472132 
.471715 

23 
22 
21 

40 
41 

42 

9.505234 
.505608 
.505981 

6.23 
6.22 

9.976532 
.976489 
.976446 

.71 
.71 

9.528702 
.529119 
.529535 

6.94 
6.94 

10.471298 
.470881 
.470465 

20 
19 
18 

43 
44 
45 

46 
47 

.506354 
.506727 
.507099 
.507471 
.507843 

e!2i 

6.21 
6.20 
6.19 

.976404 
.976361 
.976318 
.976275 
.976232 

'.71 
.71 

.72 
.72 

.529950 
.530366 
.530781 
.531196 
.531611 

6.93 
6.93 
6.92 
6.91 
6.91 

.470050 
.469634 
.469219 
.468804 
.468389 

17 
16 
15 
14 
13 

48 
49 

.508214 
.508585 

6.  19 
6.18 
6.18 

.976189 
.976146 

.72 
.72 

.72 

.532025 
.532439 

8.90 
6.90 
6.89 

.467975 
.467561 

12 
11 

50 

9.508956 

617 

9.976103 

9.532853 

10.467147 

10 

51 
52 

.509326 
.509696 

•  li 

6.16 

.976060 
.976017 

.72 
.72 

.533266 
.533679 

6.89 
6.88 

.466734 
.466321 

9 
8 

53 
54 

55 

.510065 
.510434 
.510803 

6.16 
6.15 
6.15 
6  14 

.975974 
.975930 

.975887 

.72 
.72 

.72 

ITO 

.534092 
.534504 
.534916 

6.88 
6.87 
6.87 

,465908 
.465496 
.465084 

7 
6 
5 

56 
57 
58 
59 
60 

.511172 
.511540 
.511907 
.512275 
.512642 

e!i4 

6.13 
6.12 
6.12 

.975844 
.975800 
.975757 
.975714 
.975670 

•  i  — 

.72 
.72 
.72 
.72 

.535328 
.535739 
.536150 
.536561 
.536972 

eise 

6.85 
6.85 
6.84 

.464672 
.464261 
.463850 
.463439 

.463028 

4 

3 
2 

1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

108° 


71' 


TABLE  IV.      LOGARITHMIC  SINES,  ETC.  59 

160* 


M. 

Sine. 

D.  1  '. 

Cosine. 

D.I  . 

Tang. 

D.  l". 

Cotang. 

M. 

0 

9.512642 

6-f  •« 

9.975670 

7O 

9.536972 

10.463028 

60 

1 

.513009 

.11 

6-1  1 

.975627 

.7o 

*TO 

.537382 

6QO 

.462618 

59 

2 

.513375 

.  11 

61  A 

.975583 

.  Io 

170 

.537791 

.OO 
6Qr» 

.462209 

58 

3 

.513741 

.  1U 
6AO 

.975539 

.  IO 

79 

.538202 

.00 

6QO 

.461798 

57 

4 

.514107 

.uy 

6AQ 

.975496 

•7o 

7Q 

.538611 

.82 

.461389 

56 

5 

.514472 

.uy 

6AQ 

.975452 

•  IO 

170 

.539020 

6Q1 

.460980 

55 

6 

.514837 

.Uo 

6  Aft 

.975408 

•  IO 

70 

.539429 

.ol 

601 

.460571 

54 

7 

.515202 

.Uo 
6A7 

.975365 

.  IO 

fTft 

.539837 

.ol 

6QA 

.460163 

53 

8 

.515566 

.UY 
6A7 

.975321 

•  Io 

75 

.540245 

.  oU 

6QA 

.459755 

52 

9 

.515930 

.U7 

6.06 

.975277 

•  Io 

.73 

.540653 

.oU 
6.79 

.459347 

51 

10 

9.516294 

6AK 

9.975233 

M 

9.541061 

10.458939 

50 

11 

.516657 

.UO 
6  AC 

.975189 

"70 

.541468 

6.79 
670 

.458532 

49 

12 

.517020 

.UO 

.975145 

.  IO 

.541875 

.  IO 

6rJQ 

.458125 

48 

13 
14 

.517382 
.517745 

6.'  04 

6AQ 

.975101 
.975057 

.'73 

70 

.542281 
.542688 

.  Io 

6.77 

f*  17*7 

.457719 
.457312 

47 
46 

15 

16 
17 
18 
19 

.518107 
.518468 
.518829 
.519190 
.519551 

.Uo 

6.03 
6.02 
6.02 
6.01 
6.00 

.975013 
.974969 
.974925 
.974880 
.974836 

•  IO 

.74 

.74 
.74 
.74 

.74 

.543094 
.543499 
.543905 
.544310 
.544715 

0  .  1  i 

6.76 
6.76 
6.75 
6.75 
6.74 

.456906 
.456501 
.456095 
.455690 
.455285 

45 
44 
43 

42 
41 

20 
21 

9.519911 
.520271 

6.00 

9.974792 
.974748 

.74 

9.545119 
.545524 

6.74 

6  TO 

10.454881 
.454476 

40 
39 

22 
23 

.520631 
.520990 

5  99 
5.99 

5  no 

.974703 
.974659 

!?4 

.545928 
.546331 

.  7o 
6.73 

6  TO 

.454072 
.453669 

38 
37 

24 

.521349 

.yo 

.974614 

.546735 

.14 

.453265 

36 

25 

.521707 

5.98 

.974570 

.547138 

6.72 

6T1 

.452862 

35 

26 

.522066 

rvr 

.974525 

: 

.547540 

.  ll 

6*71 

.452460 

34 

27 
28 

.522424 
.522781 

5.97 
5.96 

5QK 

.974481 
.974436 

; 

.74 

.547943 
.548345 

.71 
6.70 

67A 

.452057 
.451655 

33 
32 

29 

.523138 

.yo 
5.95 

.974391 

1 

.75 

.548747 

.  lU 

6.69 

.451253 

31 

30 

9.523495 

5/VJ 

9.974347 

ITK 

9.549149 

6/»Q 

10.450851 

30 

31 

.523852 

.y* 

.974302 

.  lO 

.549550 

.D» 

.450450 

29 

32 

.524208 

5.94 

.974257 

*7* 

.549951 

6.68 

6£O 

.450049 

28 

33 

.524564 

Q0 

.974212 

.  Io 

*TK 

.550352 

.DO 

6  AT 

.449648 

27 

34 
35 
36 

.524920 
.525275 
.525630 

5^92 
5.92 

K   ft] 

.974167 
.974122 
.974077 

.  *O 

.75 
.75 

.550752 
.551152 
.551552 

.Ol 

6.67 
6.67 

.449248 
.448843 
.448448 

26 
25 
24 

37 

.525984 

o.yi 

-   .  u  i 

.974032 

*7* 

.551952 

ft'fifi 

.448048 

23 

38 
39 

.526339 
.526693 

o.yu 
5.90 
5.89 

.973987 
.973942 

•  Io 

.75 
.75 

.552351 
.552750 

6^65 
6.65 

.447649 
.447250 

22 
21 

40 

9.527046 

9.973897 

7K 

9.553149 

fi  ft4 

10.446851 

20 

41 

.527400 

500 

.973852 

•  4U 

.553548 

6mA 

.446452 

19 

42 

.527753 

.00 

5QO 

.973807 

*7fc 

.553946 

.54 
6/50 

.446054 

18 

43 
44 

.528105 
.528458 

.00 

5.87 

507 

.973761 
.973716 

!?5 

7fi 

.554344 
.554741 

.Do 
6.63 

.445656 
.445259 

17 
16 

45 

.528810 

.01 

6   Off 

.973671 

•  ID 

.555139 

6.62 

.444861 

15 

46 

.529161 

.00 
5OJ! 

.973625 

Tfi 

.555536 

6.62 

444464 

14 

47 

.529513 

.OD 

5oe 

.973580 

.76 

7<t 

.555933 

6.61 

6M 

1444067 

13 

48 

.529864 

.  oO 

5QC 

.973535 

•  ID 

.556329 

.Ol 

.443671 

12 

49 

.530215 

.  OO 

5.84 

.973489 

!?6 

.556725 

6.60 
6.60 

.443275 

11 

50 

9.530565 

500 

9.973444 

9.557121 

6RA 

10.442879 

10 

51 

52 

.530915 
.531265 

.00 

5.83 

5  OO 

.973398 
.973352 

•I6 

.557517 
.557913 

.59 
6.59 

A   Crt 

.442483 
.442087 

9 
8 

53 
54 

.531614 
.531963 

•mm 

5.82 

E  01 

.973307 
.973261 

'.76 

7ft 

.558308 
.558702 

b.oy 
6.58 

/»  KQ 

.441692 
.441298 

7 
6 

55 
56 
57 
58 
59 
60 

.532312 
.532661 
.533009 
.533357 
.533704 
.534052 

O.ol 

5.81 
5.80 
5.80 
5.79 
5.79 

.973215 
.973169 
.973124 
.973078 
.973032 
.972986 

•  ID 

.76 
.76 
.76 

.77 
.77 

.559097 
.559491 
.559885 
.560279 
.560673 
.561066 

o.Oo 
6.57 
6.57 
6.56 
6.56 
6.55 

.440903 
.440509 
.440115 
.439721 
.439327 
.438934 

5 

4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D  1". 

Co  tang. 

D.  1  , 

Tang. 

M. 

1093 


70° 


60             TABLE    IV.      LOGARITHMIC   SINES,    ETC. 
20°                                                                                                        1  59* 

M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I  . 

Cotang. 

M. 

0 

9.534052 

57Q 

9.972986 

77 

9.561066 

6KK 

10.438934 

60 

1 

.534399 

.  4O 

57G 

.972940 

•  it 

77 

.561459 

.00 

.438541 

59 

2 
3 

.534745 
.535092 

*  4o 

5.77 

577 

.972894 
.972848 

.44 

.77 

77 

.561851 
.562244 

6^54 

6CJ. 

.438149 
.437756 

58 
57 

4 
5 

.535438 
.535783 

•  4  i 

5.76 

.972802 
.972755 

.44 

.77 
77 

.562636 
.563028 

.04 

6.53 

6     CO 

.437364 
.436972 

56 
55 

6 

.536129 

3"  25 

.972709 

.44 

77 

.563419 

.Oo 

.436581 

54 

7 
8 

.536474 
.536818 

5!  75 

67/1 

.972663 
.972617 

.44 

.77 

77 

.563811 
.564202 

e!52 

6K1 

.436189 
.435798 

53 
52 

9 

.537163 

.  44 

5.74 

.972570 

.44 

.77 

.564592 

.01 

6.51 

.435408 

51 

10 

9.537507 

57°. 

9.972524 

77 

9.564983 

6KA 

10.435017 

50 

11 

.537851 

.  4o 

570 

.972478 

.44 

77 

.565373 

.OU 

.434627 

49 

12 

.538194 

.  4O 

579 

.972431 

.44 

70 

.565763 

5*52 

.434237 

48 

13 

.538538 

.  i  £ 

571 

.972385 

•  4O 

7ft 

.566153 

5*2|j 

.433847 

47 

14 

.538880 

.  4  1 

571 

.972338 

•  4o 

7Q 

.566542 

A 

.433458 

46 

15 

.539223 

.  4  1 

57A 

.972291 

•  40 
70 

.566932 

6   Aft 

.433068 

45 

16 

.539565 

.  4U 
57A 

.972245 

.  4o 

.567320 

.4o 

.432680 

44 

17 

.539907 

.  4U 

K     fiQ 

.972198 

70 

.567709 

6*4.7 

.432291 

43 

18 

.540249 

O.O«7 

5AQ 

.972151 

•  45 

7ft 

.568098 

.44 

6/17 

.431902 

42 

19 

.540590 

.0*7 

5.68 

.972105 

.  4O 

.78 

.568486 

.44 

6.46 

.431514 

41 

20 
21 
22 
23 

9.540931 
.541272 
.541613 
.541953 

5.68 
5.67 
5.67 

9.972058 
.972011 
.971964 
.971917 

.78 
.78 
.78 

7ft 

9.568873 
.569261 
.569648 
.570035 

6.46 
6.46 
6.45 

6AK. 

10.431127 
.430739 
.430352 
.429965 

40 
39 
38 
37 

24 

.542293 

5*25 

.971870 

•  4O 
7Q 

.570422 

.40 

6    A  A 

.429578 

36 

25 
i  26 

27 

.542632 
.542971 
.543310 

6.65 
5.65 

5f*A 

.971823 
.971776 
.971729 

.40 
.78 

.78 

7Q 

.570809 
.571195 
.571581 

.44 
6.44 
6.43 

6,40 

.429191 
.428805 
.428419 

35 
34 
33 

28 
29 

.543649 
.543987 

.04 

5.64 
5.63 

.971682 
.971635 

•  (  •' 

.79 

.79 

.571967 
.572352 

.4o 

6.43 
6.42 

.428033 
.427648 

32 
31 

30 

9.544325 

5/»0 

9.971588 

7Q 

9.572738 

fi  49 

10.427262 

30 

31 

.544663 

.00 

K    gO 

.971540 

•  <  •' 
7Q 

.573123 

6J.1 

.426877 

29 

32 

.545000 

.971493 

.  4*7 

7Q 

.573507 

.41 
6/11 

.426493 

28 

33 
34 

.545338 
.545674 

6.61 

5fi1 

.971446 
.971398 

.  4*7 

.79 

7Q 

.573892 
.574276 

.41 

6.40 

6Af\ 

.426108 
.425724 

27 
26 

35 

.546011 

.01 

.971351 

•  4*7 

7Q 

.574660 

.4U 

6A(\ 

.425340 

25 

36 

.546347 

eftrt 

.971303 

•  4  *7 

.575044 

.4U 

6OQ 

'.424956 

24 

37 

.546683 

.OU 

5CQ 

.971256 

.575427 

.oil 

.424573 

23 

38 

.547019 

.0*7 

.971208 

.575810 

6.3J 

.424190 

22 

39 

.547354 

5.59 
5.58 

.971161 

.79 

.576193 

6.38 
6.38 

.423807 

21 

40 

9.547689 

5  58 

9.971113 

9.576576 

607 

10.423424 

20 

41 

.548024 

5  ^7 

.971066 

QA 

.576958 

.04 

607 

.423041 

19 

42 

.548359 

.  04 

BK.7 

.971018 

•  oU 

QA 

.577341 

.04 
607 

.422659 

18 

43 
44 

.548693 
.549027 

.04 

5.56 

t     fff* 

.970970 
.970922 

•  oU 

,80 

QA 

.577723 
.578104 

.o7 
6.36 

6QA 

.422277 
.421896 

17 
16 

45 

.549360 

O.Do 

6KK 

.970874 

.oU 

Qrt 

.578486 

.OU 
6     OK 

.421514 

15 

46 
47 

.549693 
..550026 

.00 

5.55 

5CK 

.970827 
.970779 

•  oU 

.80 

QA 

.578867 
.579248 

.oO 

6.35 

60yfl 

.421133 
.420752 

14 
13 

48 

.550359 

.00 

5KA 

.970731 

.oU 

QA 

.579629 

.o4 

6QJ. 

.420371 

12 

49 

.550692 

.04 

5.54 

.970683 

•  oU 

.80 

.580009 

.o4 

6.34 

.419991 

11 

60 

9.551024 

5KQ 

9.970635 

9.580389 

6QQ 

10.419611 

10 

51 

.551356 

.OO 
BKO 

.970586 

QA 

.580769 

•  OO 

600 

.419231 

9 

52 

.551687 

.Oo 

.970538 

•  oU 

.581149 

.00 

.418851 

8 

53 

.552018 

5.52 

.970490 

QA 

.581528 

00 

.418472 

7 

54 

.552349 

5e-| 

.970442 

•<?U 

.581907 

6.oJ 

.418093 

6 

55 

.552680 

.01 

BK-t 

.970394 

Q1 

.582286 

6.32 

6Q1 

.417714 

5 

56 
57 
58 

.553010 
.553341 
.553670 

.01 

5.50 
5.50 

5A(\ 

.970345 
.970297 
.970249 

•  ol 

.81 

.81 

01 

.582665 
.583043 
.583422 

.ol 
6.31 
b.30 

.417335 
.416957 
.416578 

4 
3 

2 

59 
60 

.554000 
.554329 

.  4*7 

5.49 

.970200 
.970152 

•  ol 

.81 

.583800 
.584177 

6'.30 

.416200 
.415823 

1 
0 

MT 

Cosine. 

D.I  . 

Sine. 

D.I    . 

Cotang. 

D.I". 

Tang. 

M. 

110C 


TABLE  IV.   LOGARITHMIC  SINES,  ETC.       61 

ar                              158° 

M. 

Sine. 

D.l  . 

Cosine. 

D.I".  Tang.   D.I". 

Cotang. 

M. 

0 

9.554329 

5.48 

9.970152 

.81 

9.584177 

6.29 

0.415823 

60 

1 

.554658 

5.48 

.970103 

.81 

.584555 

6.29 

.415445 

59 

2 

.554987 

5.47 

.970055 

.81 

.584932 

6.28 

.415068 

68 

3 

.555315 

5.47 

.970006 

.81 

.585309 

6.28 

.414691 

57 

4 

.555643 

5.46 

.969957 

.81 

.585686 

6.28 

.414314 

56 

5 

.555971 

5.46 

.969909 

.81 

.586062 

6.27 

.413938 

65 

6 

.556299 

5.45 

.969860 

.81 

.586439 

6.27 

.413561 

54 

7 

.556626 

5.45 

.969811 

.81 

.586815 

6.26 

.413185 

53 

8 

.556953 

5.44 

.969762 

.81 

.587190 

8.26 

.412810 

52 

9 

.557280 

5.44 

.969714 

.81 

.587566 

6.26 

.412434 

51 

10 

9.557606 

5.44 

9.969665 

.82 

9.587941 

6.25 

0.412059 

50 

11 

.557932 

5.43 

•969616 

.82 

.588316 

6.25 

.411684 

49 

12 

.558258 

6.43 

.969567 

.82 

.588691 

6.24 

.411309 

48 

13   .558583 

5.42 

.969518 

.82 

.589066 

6.24 

.410934 

47 

14  !  .558909 

6.42 

.969469 

.82 

.589440 

6.24 

.410560 

46 

15 

.559234 

6.41 

.969420 

.82 

.589814 

6.23 

.410186 

45 

16 

.559558 

5.41 

.969370 

.82 

.590188 

6.23 

.409812 

44 

17 

.559883 

5.40, 

.969321 

.82 

.590562 

6.22 

.409438 

43 

18 

.660207 

5.40 

.969272 

.82 

.590935 

6.22 

.409065 

42 

19 

.560531 

5.39 

.969223 

.82 

.591308 

6.22 

.408692 

41 

20 

9.560855 

6.39 

9.969173 

.82 

9.591681 

6.21 

0.408319 

40 

21 

.561178 

5.38 

.969124 

.82 

.592054 

6.21 

.407946 

39 

22 

.561501 

5.38 

.969075 

.82 

.592426 

6.20 

.407574 

38 

23 

.561824 

6.37 

.969025 

-.82 

.592798 

6.20 

.407202 

37 

24 

.562146 

5.37 

.968976 

.83 

.593171 

6.20 

.406829 

36 

25 

.562468 

5.37 

.968926 

.83 

.593542 

6.19 

.406458 

35 

26 

.562790 

5.36 

.968877 

.83 

.593914 

6.19 

.406086 

34 

27 

.563112 

5.36 

.968827 

.83 

.594285 

6.18 

.405715 

33 

28 

.563433 

5.35 

.968777 

.83 

.594656 

6.18 

.405344 

32 

29 

.563755 

5.35 

.968728 

.83 

.595027 

6.18 

.404973 

31 

30 

9.564075 

5.34 

9.968678 

.83 

9.595398 

6.17 

0.404602 

30 

31 

.564396 

5.34 

.968628 

.83 

.595768 

6.17 

.404232 

29 

32 

.564716 

6.33 

.968578 

.83 

.596138 

6.16 

.403862 

28 

33 

.565036 

5.33 

.968528 

.83 

.596508 

6.16 

.403492 

27 

34 

.565356 

5.32 

.968479 

.83 

.596878 

6.16 

.403122 

26 

35 

.565676 

5.32 

.968429 

.83 

.597247 

6.15 

.402753 

25 

36 

.665995 

5.32 

.968379 

.83 

.597616 

6.15 

.40-23*4 

24 

37 

.566314 

5.31 

.968329 

.83 

.597985 

6.15 

.402015 

23 

38 

.566632 

£.31 

.968278 

.84 

.598354 

6.14 

.401646 

22 

39 

.566951 

5.30 

.968228 

.84 

.598722 

6.14 

.401278 

21 

40 

9.567269 

5.30 

9.968178 

.84 

9.599091 

6.13 

10.400909 

20 

41 

.567587 

6.29 

.968128 

.84 

.599459 

6  13 

.400541 

19 

42 

.567904 

6.29 

.968078 

.84 

.599827 

6.13 

.400173 

18 

43 

.563222 

5.28 

.968027 

.84 

.600194 

6.12 

.399806 

17 

44 

.568539 

5.28 

.967977 

.84 

.600562 

6.12 

.399438 

16 

45 

.568856 

5.28 

.967927 

.84 

.600929 

6.12 

.399071 

15 

46 

.569172 

5.27 

.967876 

.84 

.601296 

6.11 

.398704 

14 

47 

.569488 

5.27 

.967826 

.84 

.601662 

6.11 

.398338 

13 

48 

.569804    5.26 

.967775 

.84 

.602029 

6.10 

.397971 

12 

49 

.570120    6.26 

.967725 

.84 

.602395 

6.10 

.397605 

11 

60 

9.570435 

5.25 

9  967674 

.84 

9.602761 

6.10 

10.397239 

10 

51 

.570751 

5.25 

.967624 

.84 

.603127 

6.09 

.396873 

9 

52 

.571066 

5.24 

.967573 

.85 

.603493 

6.09 

.396507 

8 

53 

.571380 

5.24 

.967522 

.85 

.603858 

6.09 

.396142 

7 

54 

.571695 

5.24 

.967471 

.85 

.604223 

6.08 

.395777 

6 

65 

.572009 

5.23 

.967421 

.85 

.604588 

6.08 

.395412 

5 

66 

.572323 

5.23 

.967370 

.85 

.604953 

6.07 

.395047 

4 

67 

.572636 

5.22 

.967319 

.85 

.605317 

6.07 

.394683 

3 

58 

.572950 

5.22 

.967268 

.85 

.605682 

6.07 

.394318 

2 

69 

.573263 

6.21 

.967217 

.85 

.606046 

6.06 

.393954 

1 

60 

.573575 

.967166 

.606410 

.393590 

0 

M. 

Cosine. 

D.I  . 

Sine. 

D.r 

Cotang 

D.I'. 

Tang. 

M. 

Ill 


68° 


62 


TABLE  IV.      LOGARITHMIC  SINES,  ETC. 


JM. 

Sine. 

D.l". 

Cosine. 

D.l" 

Tang. 

D.l". 

Cotang. 

M. 

0 

i 

2 
3 
4 
6 
6 
7 

9.673575 
.573888 
.574200 
.574512 
.574824 
.575136 
.575447 
.575758 

6.21 
6.20 
5.?0 
5.20 
5.19 
5.19 
5.18 
5  18 

9.967166 
.967115 
.967064 
.967013 
.966961 
.966910 
.966859 
.966808 

.85 
.85 
.85 
.85 
.85 
.85 
.86 

9.606410 

.606773 
.607137 
.607500 
.607863 

.608225 
,608588 
.608950 

6.06 

6.06 
6.05 
6.05 
6.05 
6.04 
6.04 

10.393590 
.393227 
.392863 
.392500 
.392137 
.391775 
.391412 
.391050 

60 
69 
58 
57 
56 
55 
54 
63 

8 
9 

.576069 
.£76379 

5.17 
5.17 

.966756 
.966705 

.86 
.86 

.609312 
.609674 

6.03 
6.03 

.390688 
.390326 

52 
51 

10 
11 
12 
13 

9.576689 
.576999 
.577309 
.577618 

5.17 
5.16 
6.16 

9.966653 
.966602 
.966550 
.966499 

.86 
.86 
.86 

9.610036 
.610397 
.610759 
.611120 

6.02 
6.02 
6.02 

10.389964 
.389603 
.389241 

.388880 

50 

49 

48 
47 

14 
15 
16 
17 
18 
19 

.577927 
.578236 
.578545 
.578853 
.579162 
.579470 

5.15 
5.14 
5.14 
5.14 
5.13 
5.13 

.966447 
.966395 
.966344 
.966292 
.966240 
.966188 

.86 
.86 
.86 
.86 
.86 
.86 

.611480 
.611841 
.612201 
.612561 
.612921 
.613281 

6.01 
6.01 
6.01 
6.00 
•  6.00 
6.00 
5.99 

.388520 
.388159 
.387799 
.387439 
.387079 
.386719 

46 
45 
44 
43 
42 
41 

20 
21 

9.579777 
.580085 

6.12 
6  12 

9.966136 
.966085 

.87 

9.613641 
.614000 

5.99 

10.386359 
.386000 

40 
S9 

22 

.580392 

61  1 

.966033 

.614359 

.385641 

38 

23 
24 
25 
26 
27 
28 
29 

.580699 
.581005 
.681312 
.581618 
.681924 
.582229 
,582535 

5.11 
5.11 
5.10 
5.10 
5.09 
5.09 
5.09 

.965981 
.965928 
.965876 
.965824 
.965772 
.965720 
.965668 

.87 
.87 
.87 
.87 
.87 
.87 
.87 

.614718 
.615077 
.615435 
.615793 
.616151 
.616509 
.616867 

6.98 
5.97 
5.97 
5.97 
5.96 
5.96 
6.96 

.385282 
.384923 
.384565 
.384207 
.383849 
.383491 
.383133 

37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.582840 
.583145 
.583449 
.583754 
.584058 
.584361 
.584665 
.584968 
.585272 
.585574 

5.08 
5.08 
5.07 
5.07 
5.06 
5.06 
5.06 
6.05 
5.05 
5.04 

9.965615 
.965563 
.965511 
.965458 
.965406 
.965353 
.965301 
.965248 
.965195 
.965143 

.87 
.87 
.87 
.87 
.88 
.88 
.88 
.88 
.88 
.88 

9.617224 
.617582 
.617939 
.618295 
.618652 
.619008 
.619364 
.619721 
.620076 
.620432 

5.95 
5.95 
5.95 
5.94 
5.94 
5.94 
5.93 
5.93 
5.93 
6.92 

10.382776 
.382418 
.382061 
.381705 
.381348 
.380992 
.380636 
.380280 
.379924 
.379568 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.585877 
.586179 
.686482 
.586783 
.587085 
.587386 
.587688 
.587989 
.588289 
.588590 

6.04 
5.04 
5.03 
6.03 
5.02 
5.02 
5.01 
6.01 
6.01 
6.00 

9.965090 
.965037 
.964984 
.964931 
.964879 
.964826 
.964773 
.964720 
.964666 
.964613 

.88 
.88 
.88 
.88 
.88 
.88 
.88 
.88 
.89 
.89 

9.  620787 
.621142 
.621497 
.621852 
.622207 
.622561 
.622915 
.623269 
.623623 
.623976 

6.92 
5.92 
5.91 
5.91 
5.91 
5.90 
5.90 
5.90 
5.89 
5  89 

10.379213 

.378858 
.378503 
.378148 
.377793 
.377439 
.377085 
.376731 
.376377 
.376024 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
65 

9.588890 
.589190 
.589489 
.589789 
.590088 
.590387- 

6.00 
4.99 
4.99 
4.99 
4.98 

9.964560 
.964507 
.964454 
.964400 
.964347 
.964294 

.89 
.89 
.89 
.89 
.89 

9.624330 
.624683 
.625036 
.625388 
.625741 
.626093 

5.89 
5.88 
5.88 
5.88 
5.87 

0.375670 
.375317 
.374964 
.374612 
.374259 
.373907 

10 
9 

8 
7 
6 
5 

66 
57 
68 
69 

.590686 
.590984 
.591282 
.591580 

4.97 
4.97 
4.97 

.964240 
.964187 
.964133 
.964080 

.89 
.89 
.89 

.626445 
.626797 
.627149 
.627501 

5.87 
5.87 
5.86 
5.86 

.373555 
.373203 
.372851 
.372499 

4 
3 
2 

60 

.591878 

.964026 

.89 

.627852 

5.86 

.372148 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 

112° 


67' 


TABLE  IY.      LOGARITHMIC  SINES,  ETC.  63 

156. 


M. 

Sine. 

D.I". 

Cosine. 

D.l" 

Tang. 

D.I". 

Cotang. 

M. 

0 

9.591878 

4fWJ 

9.964026 

on 

9.627852 

5QC 

10.372148 

60 

1 
2 

.592176 
.592473 

.9b 
4.95 
4  95 

.963972 
.963919 

.ey 
.89 
.90 

.628203 
.628554 

.  oO 

5.85 

5.85 

.371797 
.371446 

59 
58 

3 

.592770 

.963865 

.628905 

504 

.371095 

67 

4 

.593067 

4  04 

.963811 

on 

.629255 

.0* 
5QA 

.370745 

56 

5 

.593363 

A  f\A 

.963757 

'on 

.629606 

,<y± 

5Q4 

.370394 

55 

6 

.593659 

.963704 

»yu 

.629956 

,<yk 

.370044 

54 

7 

.593955 

4.93 

.963650 

.90 

.630306 

5.83 

5QO 

.369694 

53 

8 
9 

.594251 
.594547 

4i  93 
4.92 

.963596 
.963542 

!yO 
.90 

.630656 
.631005 

.OO 

5.83 
6.82 

.369344 
.368995 

52 
51 

10 
11 
12 
13 
14 
15 
16 

9.594842 
.595137 
.595432 
.595727 
.596021 
.596315 
.596609 

4.92 
4.91 
4.91 
4.91 
4.90 
4.90 

9  963488 
.963434 
.963379 
.963325 
.963271 
.963217 
.963163 

.90 
.90 
.90 
.90 
.90 
.90 

9.631355 
.631704 
.632053 
.632401 
.632750 
.633098 
.633447 

6.82 
5.82 
5.81 
5.81 
6.81 
5.80 

10.368645 
.368296 
.367947 
.367599 
.367250 
.366902 
.366553 

50 
49 
48 
47 
46 
45 
44 

17 

.596903 

4.89 

.963108 

.91 

.633795 

5.80 

.366205 

43 

18 

.597196 

4.89 

.963054 

.91 
01 

.634143 

57O 

.365857 

42 

19 

.597490 

4'.  88 

.962999 

mWM 

.91 

.634490 

.  ill 

5.79 

.365510 

41 

20 
21 

9.597783 
.598075 

4.88 

4QQ 

9.962945 
.962890 

.91 

01 

9.634838 
.635185 

5.79 

10.365162 
.364815 

40 
39 

22 

.598368 

.OO 

.962830 

.yi 

.635532 

R^-Q 

.364468 

38 

23 
24 
25 

.598660 
.598952 
.599244 

4.87 
4.87 
4.86 

.962781 
.962727 
.962672 

.91 
.91 
.91 

.635879 
.636226 
.636572 

.  10 

5.78 
6.78 

.364121 
.363774 
.363428 

37 
36 
35 

26 

.599536 

4.86 

4   Qf* 

.962617 

.91 

.636919 

5.77 

er  nrv 

J63081 

34 

27 
28 

.599827 
.600118 

.00 
4.85 

4oe 

.962562 
.962508 

!91 

O1 

.637265 
.637611 

D.i* 
6.77 

B7C 

.362735 
.362389 

33 
32 

29 

.600409 

.oD 

4.84 

.962453 

.yi 
.92 

.637956 

.  i  D 

6.76 

.362044 

31 

30 

9.600700 

9.962398 

9.638302 

10.361698 

30 

31 
32 
33 
34 
35 
36 
37 
38 

.600990 
.601280 
.601570 
.601860 
.602150 
.602439 
.602728 
.603017 

4.84 
4.84 
4.83 
4.83 
4.83 
4.82 
4.82 
4.81 

.962343 
.962288 
.962233 
.962178 
.962123 
.962067 
.962012 
.961957 

.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 

.638647 
.638992 
.639337 
.639682 
.640027 
.640371 
.640716 
.641060 

6.76 
5.75 
6.75 
5.75 
6.74 
6.74 
5.74 
5.73 

.361353 
.361008 
.360663 
.360318 
.359973 
.359629 
.359284 
.358940 

29 
28 
27 
26 
25 
24 
23 
22 

39 

.603305 

4.81 
4.81 

.961902 

!92 

.641404 

5.73 
6.73 

.358596 

21 

40 
41 

9.603594 
.603882 

4.80 

4o/\ 

9.961846 
.961791 

.92 

9.641747 
.642091 

5.73 

K.  TO 

10.358253 
.357909 

20 
19 

42 
43 

.604170 
.604457 

.oU 
4.79 

.961735 
.961680 

!92 

.642434 
.642777 

O.Tz 
5.72 

.357566 
.357223 

18 
17 

44 
45 

.604745 
.605032 

4.79 
4.79 

.961624 
.961569 

.93 
.93 

.643120 
.643463 

6.72 
5.71 

.356880 
.356537 

16 
15 

46 

47 
48 

.605319 
.605606 
.605892 

4.78 
4.78 
4.78 

.961513 
.961458 
.961402 

.93 
.93 
.93 

.643806 
.644148 
.644490 

6.71 
6.71 
5.70 

.356194 
.355852 
.355510 

14 
13 
12 

40 

.606179 

4.77 
4.77 

.961346 

.93 
.93 

.644832 

5.70 
5.70 

.355168 

11 

50 

9.606465 

9.961290 

9.645174 

10.354826 

10 

51 
52 
53 

.606751 
.607036 
.607322 

4.76 
4.76 
4.76 

4TK. 

.961235 
.961179 
.961123 

.93 
.93 
.93 

.645516 
.645857 
.646199 

5.69 
5.69 
5.69 

r  *>o 

.354484 
.354143 
.353801 

9 
8 
7 

54 
55 
56 
57 
58 
59 
60 

.C07607 
.607892 
.608177 
.608461 
.608745 
.609029 
.609313 

.  (O 

4.75 
4.74 
4.74 
4.74 
4.73 
4.73 

.961067 
.961011 
.960955 
.960899 
.960843 
.960786 
.960730 

.93 
.93 
.93 
.94 
.94 
.94 

.646540 
.646881 
.647222 
.647562 
.647903 
.648243 
.648583 

o.oy 
5.68 
5.68 
5.68 
5.67 
5.67 
5.67 

.353460 
.353119 
.352778 
.352438 
.352097 
.351757 
.351417 

6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.i". 

Sine. 

D.I" 

Cotang. 

D.I". 

Tang. 

M. 

113° 


66° 


64:     TABLE  IV.   LOGARITHMIC!  SINES,  ETC 

24* 

M 

Sine. 

D.I". 

Cosine. 

D.r 

Tang. 

D.I". 

Cotang 

M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

9.609313 
.609597 
.609880 
.610164 
.610447 
.610729 
.611012 
.611294 
.611576 
.611858 

4.73 
4.72 
'  4.72 
4.72 
4.71 
4.71 
4.71 
4.70 
4.70 
4.69 

9.960730 
.960674 
.960618 
.960561 
.960505 
.960448 
.960392 
.960335 
.960279 
.960222 

.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 

9.648583 
.648923 
.649263 
.649602 
.649942 
.650281 
.650620 
.650959 
.651297 
.651636 

5.67 
5.G6 
5.G6 
5.66 
5.65 
5.65 
5.  65 
5.64 
5.64 
5  64 

10.351417 
.351077 
.350737 
.350398 
.350058 
.349719 
.349380 
.349041 
.348703 
.348364 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.612140 
.612421 
.612702 
.612983 
.613264 
.613545 
.613825 
.614105 
.614385 
.614665 

4.69 
4.69 
4.68 
4.68 
4.68 
4.67 
4.67 
4.67 
4.66 
4.66 

9.960165 
.960109 
.960052 
.959995 
.959938 
.959882 
.959825 
.959768 
.959711 
.959654 

.95 
.95 
.95 
.95 
.95 
.95 
.95 
.95 
.95 
.95 

9.651974 
.652312 
.652650 
.652988 
.653326 
.653663 
.654000 
.654337 
.654674 
.655011 

5.64 
5.63 
5.63 
5.63 
5.62 
5.62 
5.62 
5.62 
5.61 
5  61 

10.348026 
.347688 
.347350 
.347012 
.346674 
.346337 
.346000 
.345663 
.345326 
.344989 

50 
49 
"48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.614944 
.615223 
.615502 
.615781 
.616060 
.616338 
.616616 
.616894 
.617172 
.617450 

4.65 
4.65 
4.65 
4.64 
4.64 
4.64 
4.63 
4.63 
4.63 
4.62 

9.959596 
.959539 
.959482 
.959425 
.959368 
.959310 
.959253 
.959195 
.959138 
.959081 

.95 
.95 
.95 
.95 
.96 
.96 
.96 
.96 
.96 
.96 

9.655348 
.655684 
.656020 
.656356 
.656692 
.657028 
.657364 
.657699 
.658034 
.658369 

5.61 
5.61 
5.60 
5.60 
5.60 
5.59 
5.59 
5.59 
5.58 
5  58 

0.344652 
.344316 
.343980 
.343644 
.343308 
.342972 
.342636 
.342301 
.341966 
.341631 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.617727 
.618004 
.618281 
.618558 
.618834 
.619110 
.619386 
.619662 
.619938 
.620213 

4.62 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.59 
4.59 
4.59 

9.959023 
.958965 
.958908 
.958850 
.958792 
.958734 
.958677 
.958619 
.958561 
.958503 

.96 
.96 
.96 
.96 
.96 
.96 
.96 
.97 
.97 
.97 

9.658704 
.659039 
.659373 
.659708 
.660042 
.660376 
.660710 
.661043 
.661377 
.661710 

5.58 
5.58 
5.57 
5.57 
5.57 
5.56 
5.56 
5.56 
5.56 
5  55 

0.341296 
.340961 
.340627 
.340292 
.339958 
.339624 
.339290 
.338957 
.338623 
.338290 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.620488 
.620763 
.621038 
.621313 
.621587 
.621861 
.622135 
.622409 
.622682 
.622956 

4.58 
4.58 
4.58 
4.57 
4.57 
4.57 
4.56 
4.56 
4.56 
4.55 

9.958445 
.958387 
.958329 
.958271 
.958213 
.958154 
.958096 
.958038 
.957979 
.957921 

.97 

.97 
.97 
.97 
.97 
.97 
.97 
.97 
.97 
.97 

9.662043 
.662376 
.662709 
.663042 
.663375 
.663707 
.664039 
.664371 
.664703 
.665035 

5.55 
5.55 
5.54 
5.54 
5.54 
5.54 
5.53 
5.53 
5.53 
5  53 

0.337957 
.337624 
.337291 
.336958 
.336625 
.336293 
.335961 
.335629 
.335297 
.334965 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
51 
52 
53 
54 

9.623229 
.623502 
.623774 
.624047 
.624319 

4.55 
4.54 
4.54 
4.54 

9.957863 
.957804 
.957746 
.957687 
.957628 

.97 

.98 
.98 
.98 

9.665366 
.665697 
.666029 
.666360 
.666691 

5.52 
5.52 
5.52 
5.51 

0.334634 
.334303 
.333971 
.333640 
:  333309 

10 
9 
8 

6 

55 

.624591 

4KO 

.957570 

.667021 

5.51 

.332979 

5 

56 
57 
58 
59 
60 

.624863 
.625135 
.625406 
.625677 
.625948 

4.53 
4.52 
4.52 
4.52 

.957511 
.957452 
.957393 
.957335 
.957276 

.98 
.98 
.98 
.98 

.667352 
.667682 
.668013 
.668343 
.668672 

5.51 
5.51 
5.50 
5.50 
5.50 

.332648 
.332318 
.331987 
.331657 
.331328 

4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

114 


65° 


TABLE   IV.      LOGARITHMIC   SINES,   ETC. 


65 


25 


M. 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

9.625948 

4*1 

9.957276 

QQ 

9.668673 

K  KA 

10.  331327 

60 

1 
2 

.626219 
.626490 

.01 

4.51 

4K| 

.957217 
.957158 

.yo 
.98 

.669002 
.669332 

O.Ov 

5.49 

.330998 
.330668 

59 
58 

3 
4 

.626760 
.627030 

.Ol 

4.50 

4trt 

.957099 
.957040 

^98 

QQ 

.669661 
.669991 

5\49 

K  A(\ 

.330339 
.330009 

67 
66 

5 
6 

7 

.627300 
.627570 
.627840 

.DU 

4.50 
4.49 

4AQ 

.956981 
.956921 
.956862 

.yy 
.99 
.99 

QQ 

.670320 
.670649 
.670977 

5i48 
5.48 

5AQ. 

.329680 
.329351 
.329023 

65 
64 
53 

8 

.628109 

•*y 
4  49 

.956803 

.yy 

QQ 

.671306 

«4o 

547 

.328694 

62 

9 

.628378 

4i48 

.956744 

«yy 
.99 

.671634 

•  4i 
5.47 

.328366 

61 

10 
11 

9.628647 
.628916 

4.48 

4JQ 

9.956684 
.956625 

.99 

QQ 

9.671963 
.672291 

5.47 

547 

10.328037 
.327709 

50 
49 

12 

.629185 

,*O 

4A7 

.956566 

.  yy 

QQ 

.672619 

•  *'  t 

54fS 

.327381 

48 

13 

.629453 

•ffl 

4  AT 

,956506 

.yy 

.672947 

•  4O 
5  Aft 

.327053 

47 

14 

.629721 

.41 

447 

.956447 

.99 

QQ 

.673274 

.TO 
5AR 

.326726 

46 

15 

.629989 

•  4f 

4  46 

.956387 

.  yy 

QQ 

.673602 

•  4O 
54f* 

.326398 

45 

16 
17 

.630257 
.630524 

4'.46 

.956327 
.956268 

.yy 
.99 

QQ 

.673929 
.674257 

•  4O 

5.45 
5  45 

.326071 
.325743 

44 
43 

18 
19 

.630792 
i  631059 

4*.45 
4.45 

.956208 
.956148 

.yy 
.00 
.00 

.674584 
.674910 

5*45 
5.45 

.325416 
.325090 

42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
•29 

9.631326 
.631593 
.631859 
.632125 
.632392 
.632658 
.632923 
.633189 
.633454 
.633719 

4.45 
4.44 
4.44 
4.44 
4.43 
4.43 
4.43 
4.42 
4.42 
4.42 

9.956089 
.956029 
.955969 
.955909 
.955849 
.955789 
.955729 
.955669 
•955609 
.955548 

.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 

9.675237 
.675564 
.675890 
.676216 
.676543 
.676869 
.677194 
.677520 
.677846 
.678171 

5.44 
5.44 
5.44 
5.44 
5.43 
5.43 
5.43 
5.42 
5.42 
5.42 

10.324763 

.32443G 
.324110 
.323784 
.323457 
.323131 
.322806 
.322480 
.322164 
.321829 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 

9.633984 
.634249 
.634514 
.634778 

4.41 
4.41 
4.41 
4  40 

9.955488 
.955428 
.955368 
.955307 

.00 
.01 
.01 
Ol 

9.678496 
.678821 
.679146 
.679471 

5.42 
5  41 
5.41 

K  Al 

10.321504 
.321179 

.320854 
.320529 

30 

29 
28 
27 

34 

35 
36 
37 

.635042 
.635306 
.635570 
.635834 

4^40 
4.40 
4.39 

40Q 

.955247 
.955186 
.955126 
.955065 

.Ul 

.01 
.01 
.01 

A1 

.679795 
.680120 
.680444 
.680768 

O.41 

5.41 
6.40 
5.40 

5Af\ 

.320205 
.319880 
.319556 
.319232 

26 
25 
24 
23 

38 

.636097 

.«>y 

40Q 

.955005 

.Ul 
01 

.681092 

.4U 
5Af\ 

.318908 

22 

39 

.636360 

.O«7 

4.38 

.954944 

.Ul 

.01 

.681416 

•  4U 

5.39 

.318584 

21 

40 

9.636623 

40Q 

9.954883 

A1 

9.681740 

10.318260 

20 

41 

.636886 

.OO 

.954823 

.01 

A1 

.682063 

R*M 

.317937 

19 

42 

.637148 

4  Q7 

.954762 

.01 

A1 

.682387 

50Q 

.317613 

18 

43 

.637411 

.Of 

407 

.954701 

.Ul 

Al 

.682710 

.oy 

500 

.317290 

17 

44 

.637673 

.04 
4Q7 

.954640 

.Ul 

AO 

.683033 

.  OO 
500 

.316967 

16 

45 

.637935 

•  Ol 

.954579 

,U<fi 

AO 

.683356 

.OO 

50Q 

.316644 

15 

46 

.638197 

AM 

.954518 

.uz 

.683679 

.OO 
600 

.316321 

14 

47 
48 
49 

.638458 
.638720 
.638981 

4l36 
4.35 
4.35 

.954457 
.954396 
.954335 

!02 
.02 
.02 

.684001 
.684324 
.684646 

.OO 

6.37 
5.37 
5.37 

.315999 
.315676 
.315354 

13 
12 
11 

50 
6! 
52 
63 

9.639242 
.639503 
.639764 
.640024 

4.35 
4.34 
4.34 

A  O£ 

9.954274 
.954213 
.954152 
.954090 

.02 
.02 
.02 

AO 

9.684968 
.685290 
.685612 
.685934 

6.37 
5.36 
5.36 

K  Q^S 

10.315032 
.314710 
.314388 
.314066 

10 
9 

8 
7 

64 

.640284 

A  OQ 

.954029 

bVH 

.686255 

o.oo 

.313745 

6 

55 
56 
67 
58 

s 

.640544 
.640804 
.641064 
.641324 
.641583 
.641842 

4.OO 

4.33 
4.33 
4.32 
4.32 
4.32 

.953968 
.953906 
.953845 
.953783 
.953722 
.953660 

.02 
.02 
.02 
.03 
.03 
.03 

.686577 
.686898 
.687219 
.687540 
.687861 
.688182 

6.36 
5.35 
5.35 
5.35 
5.35 
5.35 

.313423 
.313102 
.312781 
.312460 
.312139 
.311818 

5 

4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

^L 

115 


64* 


66 


TABLE    IV.      LOGARITHMIC    SINES,    ETC. 


153° 


M 

Sine. 

D.I". 

Cosine. 

D.I" 

.  Tang. 

D.l  ••. 

Cotang 

M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

9.641842 
.642101 
.642360 
.642618 
.642877 
.643135 
.643393 
.643650 
.643908 
.644165 

4.32 
4.31 
4.31 
4.31 
4.30 
4.30 
4.30 
4.29 
4.29 
4.29 

9.953660 
.953599 
.933537 
.953475 
.953413 
.95:{352 
.953290 
.QB32SS 
.953166 
.953104 

1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1  03 
1.03 

9.688182 
.  688502 
.688823 
.689143 
.680463 
.689783 
.690103 
.690423 
.690742 
.691062 

5.34 
5.34 
5.34 
5.34 
5.33 
5.33 
5.33 
5.33 
5.32 
5.32 

10.311818 
.311498 
.311177 
.310857 
.310537 
.310217 
.309897 
.309577 
.309258 
.308938 

60 
59 

58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.644423 
-.644680 
.644936 
.645193 
.645450 
.645706 
.645962 
.646218 
.646474 
.646729 

4.28 
4.28 
4.28 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

9.953042 
.952980 
.952918 
.952855 
.952793 
.952731 
.952669 
.952606 
.952544 
.952481 

1.03 
1.04 
1.04 
1.04 
1.04 
1.04 
±.04 
1.04 
1.04 
1.04 

5.691381 
.691700 
.692019 
.692338 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 

5.32 
5.32 
5.31 
5.31 
5.31 
5.31 
5.30 
5.30 
5.30 
5.30 

10.308619 
.308300 
.307981 
.307662 
.307344 
.307025 
.306707 
.306388 
.306070 
.305752 

BO 
49 
-48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.646984 
.647240 
.647494 
.647749 
.648004 
.648258 
.648512 
.648766 
.649020 
.649274 

4.25 
4.25 
4.25 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

9.952419 
.952356 
.952294 
.952231 
.9521G8 
.952106 
.952043 
.951980 
.951917 
.951854 

1.04 
1.04 
1.04 
1.04 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 

9.694566 
.694883 
.695201 
.695518 
.695830 
.696153 
.696470 
.696787 
.697103 
.697420 

5.29 
5.29 
5.29 
5.29 
5.29 
5.28 
5.28 
5.28 
5.28 
5.27 

10.305434 
.305117 
.304799 
.304482 
.304164 
.303847 
.303530 
.303213 
.302897 
.302580 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.649527 
.649781 
.650034 
.650287 
.650539 
.650792 
.651044 

4.22 
4.22 
4.22 
4.21 
4.21 
4.21 

9.951791 
.951728 
.951665 
.951602 
.951539 
.951476 
.951412 

1.05 
1.05 
1.05 
1.05 
1.05 
1.05 

9.697736 
.698053 
.698369 
.698685 
.699001 
.699316 
.699632 

5.27 
5.27 
5.27 
5.26 
5.26 
5.26 

0.302264 
.301947 
.301631 
.301315 
.300999 
.300684 
.300368 

30 

29 
28 
27 
26 
25 
24 

37 
38 
39 

.651297 
.651549 
.651800 

4.20 
4.20 
4.19 

.951349 
.951286 
.951222 

1  06 
1.06 
1.06 

.699947 
.700263 
.700578 

5.26 
5.26 
5.25 
5.25 

.300053 
.299737 
299422 

23 
22 
21 

40 
41 
42 
43 
44 

9.652052 
.652304 
.652555 
.652806 
.653057 

4  19 
4.19 
4.18 
4.18 

4  -JO 

9.951159 
.951096 
.951032 
.950968 
.950905 

1.06 
1.06 
1.06 
1.06 

9.700893 
.701208 
,701523 
.701837 
.702152 

5.25 
5.25 
5.24 
5.24 

0.299107 
.298792 
.298477 
.298163 
.297848 

20 
19 
18 
17 
16 

45 
46 
47 
48 
49 

.653308 
.653558 
.653808 
.654059 
.654309 

4.18 
4.17 
4.17 
4.17 
4.16 

.950841 
.950778 
.950714 
.950650 
.950586 

1.06 
1.06 
1.06 
1.06 
1.06 

.702466 
.702781 
.703095 
.703409 
.703723 

5.24 
5.24 
5.23 
5.23 
5.23 

.297534 
.297219 
.296905 
.296591 
.296273 

15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 

9.654558 
.654808 
.655058 
.655307 
.655556 
.655805 
.656054 
.656302 
.656551 

4.16 
4.16 
4.15 
4.15 
4.15 
4.15 
4.14 
4.14 

9.950522 
.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 

9.704036 
.704350 
.704663 
.704977 
.705290 
.705603 
705916 
.706228 
.706541 

5.23 
5.22 
5.22 
5.22 
5.22 
5.22 
5.21 
5.21 

0.295964 
.295650 
.295337 
.295023 
.294710 
.294397 
.204084 
.293772 
.293459 

10 
9 
8 
7 
6 
5 
4 
3 
2 

59 
60 

.656799 
.657047 

4.13 

.949945 
.949881 

1.07 

.706854 
.707166 

5.21 
5.21 

.293146 
.292834 

1 
0 

M. 

Cosine. 

D.I  . 

Sine. 

D.I". 

Cotang.  D.I". 

Tang. 

M. 

116C 


27C 


TABLE  IV.      LOGARITHMIC  SINES,  ETC.  67 

152* 


M. 

Sine. 

D.I". 

Cosine. 

D.I  . 

Tang. 

D.I". 

Cotang. 

M. 

0 

2 
3 

9.657047 
.657295 
.657542 
.657790 

.13 
.13 

.12 

.   -|f> 

9.949881 
.949816 
.949752 
.949688 

1.07 
1.07 
1.07 

IAQ 

9.707166 
.707478 
.707790 
.708102 

5.20 
5.20 
5.20 

5  on 

10.292834 
.292522 
.292210 

.291898 

60 
69 

68 
57 

4 
5 
6 

.658037 
.658284 
.658531 

.  1*5 

.12 
.12 

.949623 
.949558 
.949494 

.Uo 

1.08 
1.08 

.708414 
.708726 
.709037 

.Aj 

5.20 
6.19 

5  1O 

.291586 
.291274 
290963 

66 
65 
64 

.  7 
8 

.658778 
.659025 

.11 

.949429 
.949364 

lios 

.709349 
.709660 

.  1*7 

5.19 

.290651 
.290340 

63 
52 

9 

.659271 

'.  .11 
.10 

.949300 

1.08 
1.08 

.709971 

5.19 
5.18 

.290029 

51 

10 
11 

9.659517 
.659763 

.10 

9.949235 
.949170 

1.08 

IAQ 

9.710282 
-  .710593 

5.18 

51Q. 

10.289718 

.289407 

60 
49 

12 

.660009 

.949105 

.  Uo 

IAQ 

.710904 

•  lo 

51Q 

.289096 

48 

13 

.660255 

AQ 

.949040 

.Uo 

IAQ 

.711215 

.lo 

51ft 

.288785 

47 

14 

.660501 

.uy 

.948975 

.  UO 
IAQ 

.711525 

.lo 

51T 

.288475 

46 

15 

.660746 

.09 

.948910 

.Uo 

IAQ 

.711836 

•  !• 

5-ffT 

.288164 

45 

16 

.660991 

.09 

,   AQ 

.948845 

.Uo 

IAQ 

.712146 

.17 

61T 

.287854 

44 

17 

.661236 

.Uo 

.948780 

.uy 

.712456 

.14 

.287544 

-43 

18 

.661481 

.08 

>  no 

.948715 

1.09 

1  OQ 

.712766 

5.17 

517 

.287234 

42 

19 

.661726 

.Uo 

.08 

.948650 

i  .uy 
1.09 

.713076 

*  1  i 
5.16 

.286924 

41 

20 

9.661970 

9.948584 

1/Vl 

9.713386 

5flM 

10.286614 

40 

21 

22 

.662214 
.662459 

.07 

.948519 
.948454 

.uy 
1.09 

Irwi 

.713696 
.714005 

.lo 
5.16 

51/5 

.286304 
.285995 

39 
38 

23 
24 

.662703 
.662946 

!oe 

.948388 
.948323 

.uy 
1.09 

.714314 
.714624 

.  ID 
5.15 

51  K. 

.285686 
.285376 

37 
36 

25 
26 

.663190 
.663433 

4i<)6 

.948257 
.948192 

l!(» 

.714933 
.715242 

.  ID 
6.15 

.285067 
.284758 

35 
34 

27 

.663677 

4.05 

.948126 

1.09 

.715551 

5.15 

.284449 

33 

28 

.663920 

4.05 

A  A*\ 

.948060 

1.09 

IAQ 

.715860 

5.15 

511 

.284140 

32 

29 

.664163 

4.UD 

4.05 

.947995 

.uy 
1.10 

.716168 

.  1% 
5.14 

.283832 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.664406 
.664648 
.664891 
.665133 
.665375 
.665617 
.665859 
.666100 
.666342 
.666583 

4.04 
4.04 
4.04 
4.03 
4.03 
4.03 
4.03 
4.02 
4.02 
4.02 

9.947929 
.947863 
.947797 
.947731 
.947665 
.947600 
.947533 
.947467 
.947401 
.947335 

1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 

9.716477 
.716785 
.717093 
.717401 
.717709 
.718017 
.718325 
.718633 
.718940 
.719248 

5.14 
6.14 
5.14 
6.13 
5.13 
6.13 
6.13 
5.13 
6.12 
6.12 

10.283523 
.283215 
.282907 
.282599 
.282291 
.281983 
.281675 
.281367 
.281060 
.280752 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 

9.666824 
.667065 
.667305 
.667546 
.667786 
.668027 
.668267 
.668506 

4.01 
4.01 
4.01 
4.01 
4.00 
4.00 
4.00 

o  (\f) 

9.947269 
.947203 
.947136 
.947070 
.947004 
.946937 
.946871 
.946804 

1.10 

1.11 
1.11 
1.11 
1.11 
1.11 
1.11 

1-1-1 

9.719555 

.719862 
.720169 
.720476 
.720783 
.721089 
.721396 
.721702 

6.12 
6.12 
5.11 
5.11 
6.11 
6.11 
6.11 
61  f\ 

10.280445 
.280138 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 

20 
19 
18 
17 
16 
15 
14 
13 

48 
49 

.668746 
.668986 

«>.yy 
3.99 
3.99 

.946738 
.946671 

.11 

1.11 
1.11 

.722009 
.722315 

.10 
5.10 
5.10 

.277991 
.277685 

12 
11 

50 
51' 

9.669225 
.669464 

3.99 

3  no 

9.946604 
.946538 

1.11 

11  1 

9.722621 

.722927 

5.10 

51  A 

10.277379 
.277073 

10 
9 

52 

.669703 

.yo 

.946471 

.  11 

111 

.723232 

.  1U 
K  Afl 

.276768 

8 

53 
54 
55 
56 
57 
58 
59 
60 

.669942 
.670181 
.670419 
.670658 
.670896 
.671134 
.671372 
.671609 

3^98 
3.98 
3.97 
3.97 
3.97 
3.96 
3.96 

.946404 
.946337 
.946270 
.946203 
.946136 
.946069 
.946002 
.945935 

.11 

1.11 

1.12 
1.12 
1.12 
1.12 
1.12 
1.12 

.723538 
.723844 
.724149 
.724454 
.724760 
.725065 
.725370 
.725674 

o.uy 
5.09 
5.09 
5.09 
5.09 
6.08 
5.08 
5.08 

.276462 
.276156 
.275851 
.275546 
.275240 
.274935 
.274630 
.274326 

7 
6 
5 

4 
3 
2 

1 
0 

M. 

Cosine. 

D.I". 

Sine. 

Dl". 

Cotane. 

D.I". 

Tang. 

M. 

117C 

62° 

TABLE  IV.      LOGAKITHMIO   SINES,    ETC. 


151' 


M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

M. 

0 

9.671609 

9.945935 

11  O 

9.725674 

5AQ 

10.274326 

60 

1 

.671847 

3.96 

.945868 

-  \  £ 

.725979 

.Uo 

.274021 

69 

2 
3 
4 
5 
6 
7 
8 
9 

.672084 
.672321 
.672558 
.672795 
.673032 
.673268 
.673505 
.673741 

3.96 
3.95 
3.95 
3.95 
3.94 
3.94 
3.94 
3.94 
3.93 

.945800 
.945733 
.945666 
.945598 
.945531 
.945464 
.945396 
.945328 

l'.12 
1.12 
1.12 
1.12 
1.12 
1.13 
1.13 
1.13 

.726284 
.726588 
.726892 
.727197 
.727501 
.727805 
.728109 
.728412 

5.08 
5.07 
5.07 
6.07 
5.07 
5.07 
6.06 
5.06 
5.06 

.273716 
.273412 
.273108 
.272803 
.272499 
.272195 
.271891 
.271588 

58 
57 
56 
65 
64 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 

9.673977 
.674213 

.674448 
.674684 
.674919 
.675155 
.675390 

3.93 
3.93 
3.93 
3.92 
3.92 
3.92 

3  O1 

9.945261 
.945193 
.945125 
.945058 
.944990 
.944922 
.944854 

1.13 
1.13 
1.13 
1.13 
1.13 
1.13 

11  Q 

9.728716 
.729020 
.729323 
.729626 
.729929 
.730233 
.730535 

6.06 
5.06 
6.05 
5.05 
5.05 
6.05 

10.271284 
.270980 
.270677 
.270374 
.270071 
.269767 
.269465 

50 
49 
48 
47 
46 
45 
44 

17 
18 
19 

.675624 
.675859 
.676094 

.yi 
S.91 
3.91 
3.91 

.944786 
.944718 
.944650 

.  lo 
1.13 
1.13 
1.13 

.730838 
.731141 
.731444 

6.05 
5.05 
5.04 
6.04 

.269162 
.268859 
.268556 

43 
42 
41 

20 
21 
22 
23 
24 

9.676328 
.676562 
.676796 
.677030 
.677264 

3.90 
3.90 
3.90 
3.90 

3QQ 

9.944582 
.944514 
.944446 
.944377 
.944309 

1.14 
1.14 
1.14 
1.14 
1  14 

9.731746 
.732048 
.732351 
.732653 
.732955 

6.04 
6.04 
5.04 
5.04 

K  A'-i 

10.268254 
.267952 
.267649 
.267347 
.267045 

40 
39 
38 
37 
36 

25 
26 
27 
28 

.677498 
.677731 
.677964 
.678197 

.Oi7 

3.89 
3.89 
3.88 

3QQ 

.944241 
.944172 
.944104 
.944036 

l!l4 
1.14 
1.14 

11  1 

.733257 
.733558 
.733860 
.734162 

O.Uo 

6.03 
5.03 
5.03 

.266743 
.266442 
.266140 
.265838 

35 
34 
33 
32 

A 

.678430 

.OO 

3.88 

.943967 

.  14 

1.14 

.734463 

5.02 
6.02 

.265537 

31 

30 

9.678663 

3QO 

9.943899 

111 

9.734764 

10.265236 

30 

31 
32 
33 
34 
35 
36 
37 

.678895 
.679128 
.679360 
.679592 
.679824 
.680056 
.680288 

.OO 

3.87 
3.87 
3.87 
3.87 
3.86 
3.86 

3Q/> 

.943830 
.943761 
.943693 
.943624 
.943555 
.943486 
.943417 

.14 

1.14 
1.15 
1.15 
1.15 
1.15 
1.15 

11  K 

.735066 
.735367 
.735668 
.735969 
.736269 
.736570 
.736871 

5.02 
5.02 
5.02 
5.01 
5.01 
5.01 
5.01 

.264934 
.264633 
.264332 
.26403^ 
.263731 
.263430 
.263129 

29 
28 
27 
26 
25 
24 
23 

38 
39 

.680519 
.680750 

.OD 

3.86 
3.85 

.943348 
.943279 

.ID 

1.15 
1.15 

.737171 
.737471 

5.01 
6.01 
5.00 

.262829 
.262529 

22 
21 

40 
41 
42 
43 
44 
45 

9.680982 
.681213 
.681443 
.681674 
.681905 
.682135 

3.85 
3.85 
3.84 
3.84 
3.84 

3QJ. 

9.943210 
.943141 
.943072 
.943003 
.942934 
.942864 

1.15 

1.15 
1.15 
1.15 
1.15 

11  A 

9.737771 
.738071 
.738371 
.  738671 
.738971 
.739271 

5.00 
5.00 
5.00 
5.00 
4.99 

4  Art 

10.262229 
.261929 
.261629 
.261329 
.261029 
.260729 

20 
19 
18 
17 
16 
15 

46 
47 
48 

.682365 
.682595 
.682825 

.tr* 

3.83 
3.83 

3QO 

.942795 
.942726 
.942656 

.  lu 

1.16 
1.16 

.739570 
.739870 
.740169 

.yy 
4.99 
4.99 
4  no 

.260430 
.260130 
.259831 

14 
13 
12 

49 

.683055 

.OO 

3.83 

.942587 

1.16 
1.16 

.740468 

.yy 
4.98 

.259532 

11 

50 

9.683284 

9.942517 

Iic 

9.740767 

4OQ 

10.259233 

10 

51 
52 
53 
54 

.683514 
.683743 
.683972 
.684201 

3.82 
3.82 
3.82 

3Q1 

.942448 
.942378 
.942308 
.942239 

.ID 
1.16 
1.16 
1.16 

Itfi. 

.741066 
.741365 
.741664 
.741962 

.98 
4.98 
4.98 
4.98 

.258934 
.258635 
.258336 
.258038 

9 
8 
7 
6 

55 
56 

57 

.684430 
.684658 
.684887 

.  ol 

3.81 
3.81 

Sort 

.942169 
.942099 
.942029 

.  lo 
1.16 
1.16 

117 

.742261 
.742559 
.742858 

4L97 

4.97 

.257739 
.257441 
.257142 

5 
4 
3 

68 
59 

.685115 
.685343 

.Ov/ 

3.80 

O  QA 

.941959 
.941889 

•  ll 

1.17 

.743156 

.743454 

4.97 
4.97 

.256844 
.256546 

2 
1 

60 

.685571 

o.oU 

.941819 

1.17 

.743752 

4.97 

.256248 

0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

118° 


29' 


TABLE  IV.      LOGARITHMIC  SINES,  ETC.  69 

150° 


M. 


37 


Sine. 


9.685571 
.685799 


.686264 


.686709 


.687163 
.687389 
.687616 

9.687843 


.690772 
.690996 
.691220 
.691444 


.692115 


.692785 


.693231 
.693453 


.694120 


9.694564 


.695007 
.695229 


.695671 


.696334 
.696554 

9.696775 


.697215 
.697435 


.697874 


3.80 
3.79 
3.79 
3.79 
3.79 
3.78 
3.78 
3.78 
3.78 
3.77 

3.77 
3.77 
3.77 
3.76 
3.76 
3.76 
3.76 
3.75 
3.75 
3.75 

3.75 
3.74 
3.74 
3.74 
3.74 
3.73 
3.73 
3.73 
3.73 
3.72 

3.72 
3.72 
3.72 
3.71 
3.71 
3.71 
3.71 
3.70 
3.70 
3.70 

3.70 
3.69 
3.69 
3,69 
3.69 
3.68 
3.68 
3.68 
3.68 
3.67 

3.67 
3.67 
3.67 
3.66 
3.66 
3.66 
3.66 
3.65 
3.65 
3.65 


M.     Cosine.     D.  1". 

119° 

30 


Cosine.    D.I'.     Tang.      D.I".    Cotang.    M. 


9.941819 
.941749 
.941679 


.941539 


.941398 
.941328 
.941258 
.941187 

9.941117 
.941046 
.940975 

.940905 
.940834 
.940763 


.940622 

.940.V,1 


9.940409 
.940338 
.940267 


.940125 
.940054 


.939768 


.939554 
.939482 
.939410 


.939267 
.939195 
.939123 


.938703 


.938047 

.938475 
.938402 


.938258 
.938185 
.938113 


.937967 


.937749 


.937604 
.937531 


Sine. 


1.17 

1.17 
.17 

.17 

.17 
.17 
.17 
.17 
.17 
.17 

.18 
.18 
.18 
.18 
.18 
.18 
.18 
.18 
.18 
.13 

.18 
.18 
.19 

.19 
.19 
.19 
.19 

.19 
.19 
.19 

.19 
.19 
.19 
.19 
.19 
.20 
.20 
.20 
.20 
.20 

.20 
.90 
.90 

.20 
.90 
.90 

.20 
.21 
.21 
.21 

.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
1.22 


D.I' 


9.743752 
.744050 
.744348 
.744645 
.744943 
.745240 
.745538 
.745835 
.746132 
.746429 

9.746726 
.747023 
.747319 
.747616 
.747913 
.748209 
.748505 
.748801 


.749393 


.749985 
.750281 
.750576 
.750372 
.751167 
.751462 
.751757 
.752052 
.752347 

9.752642 
.752937 
.753231 
.753526 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 

9.755585 
.755878 
.756172 
.756465 
.756759 
.757052 
.757345 
.757638 
.757931 
.758224 

9.758517 
.758810 
.759102 


.759687 
.759979 
.760272 
.760564 
.760856 
.761148 
.761439 


Cotang. 


4.96 
4.96 
4.96 
4.96 
4.96 
4.96 
4.95 
4.95 
4.95 
4.95 

4.95 
4.95 
4.94 
4.94 
4.94 
4.94 
4.94 
4.93 
4.93 
4.93 

4.93 
4.93 
4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.91 

4.91 
.91 

.91 
.91 
.91 
.90 

.90 
.90 
.90 

.90 


D.r 


10.256248 
.255950 
.255652 
.255355 
.255057 
.254760 
.254462 
.254165 
.253868 
.253571 

10.253274 
.252977 

.252681 


.251791 
.251495 
.251199 

.250903 


10.250311 
.250015 
.249719 
.249424 
.249128 
.248833 
.248538 
.248243 
.247948 
.247653 

10.247358 
.247063 
.246769 
.246474 
.246180 
.245885 
.245591 
.245297 
.245003 
.244709 

10.244415 
.244122 
.243828 
.243535 
.243241 
.242948 
.242655 
.242362 
.242069 
.241776 

10.241483 
.241190 

.240898 
.240605 
.240313 
.240021 
.239728 
.239436 
.239144 
.238852 
.238561 


Tang. 


60° 


70     TABLE  IV.   LOGARITHMIC  SINES,  ETC. 

3O°                                           149' 

M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.  l". 

Cotang. 

M. 

0 

9.698970 

3/*e 

9.937531 

9.761439 

40ft 

10.238561 

60 

1 

2 
3 

.699189 
.699407 
.699626 

.DO 

3.64 
3.64 

.937458 
.937385 
.937312 

l'.22 
1.22 

.761731 
.762023 
.762314 

.OU 

4.86 
4.86 

.238269 
.237977 
.237686 

59 

58 
57 

4 
5 
6 

7 

.699844 
.700062 
.700280 
.700498 

3.64: 

3.63 
3.63 

.937238 
.937165 
.937092 
.937019 

l!22 
1.22 
1.22 

.762606 
.762897 
.763188 
.763479 

4^86 
4.85 
4.85 

4QK 

.237394 
.237103 
.236812 
.236521 

56 
55 
54 
53 

8 

.700716 

3.63 

3/50 

.936946 

1  .22 

Ion 

.763770 

.OO 
4  OK 

.236230 

52 

9 

.700933 

.DO 

3.62 

.936872 

.352 

1.22 

.764061 

.oO 

4.85 

.235939 

51 

10 
11 
12 
13 
14 
15 

9.701151 
.701368 
.701585 
.701802 
.702019 
.702236 

3.62 
3.62 
3.62 
3.61 
3.61 

3/M 

9.936799 
.936725 
.936652 
.936578 
.936505 
.936431 

1.22 
1.23 
1.23 
1.23 
1.23 

1OQ 

9:764352 
.764643 
.764933 
.765224 
,  .765514 
.765805 

4.85 

4.84 
4.84 
4.84 
4.84 
484 

10.235648 
.235357 
.235067 
.234776 
.234486 
.234195 

50 
49 

48 
47 
46 
45 

16 
17 

.702452 
.702669 

.  Ol 

3.61 

.936357 
.936284 

.Zo 

1.23 

.766095 
.766385 

.  <y± 

4.84 

400 

.233905 
.233615 

44 
43 

18 

.702885 

3£A 

.936210 

1OQ 

.766675 

.00 

400 

.233325 

42 

19 

.703101 

.OU 

3.60 

.936136 

.zo 
1.23 

.766965 

.00 

4.83 

.233035 

41 

20 
21 
22 

9.703317 
.703533 
.703749 

3.60 
3.59 

9.936062 
.935988 
.935914 

1.23 
1.23 

1OQ 

9.767255 
.767545 
.767834 

4.83 
4.83 

10.232745 
.232455 
.232166 

40 
39 

38 

23 
24 

.703964 
.704179 

3^59 

3K(\ 

.935840 
.935766 

•220 

1.23 

.768124 
.768413 

4Q<> 

.231876 
.231587 

37 
36 

25 
26 
27 
28 

.704395 
.704610 
.704825 
.705040 

-Oi/ 

3.59 
3.58 
3.58 

.935692 
.935618 
.935543 
.935469 

1^24 
1.24 
1.24 

.768700 
.768992 
.769281 
.769571 

.  oZ 

4.82 
4.82 
4.82 

4  CO 

.231297 
.231008 
.230719 
.230429 

35 
34 
33 
32 

29 

.705254 

3.58 
3.58 

.935395 

l!24 

.769860 

.  OJ& 

4.82 

.230140 

31 

80 
81 
82 
83 

9.705469 
.705683 
.705898 
.706112 

3.57 

3.57 
3.57 

3*7 

9.935320 
.935246 
.935171 
.935097 

1.24 
1.24 
1.24 
i  .24 

9.770148 
.770437 
.770726 
.771015 

4.81 
4.81 
4.81 

A  Q1 

10.229852 
.229563 
.229274 
.228985 

30 
29 
28 
27 

84 

.706326 

.Ol 

.935022 

.771303 

4.  ol 

4Qi 

.228697 

26 

85 
86 
87 
38 
39 

.706539 
.706753 
.706967 
.707180 
.707393 

3^56 
3.56 
3.56 
3.55 
3.55 

.934948 
.934873 
.934798 
.934723 
.934649 

l!24 
1.25 
1.25 
1,25 
1.25 

.771592 
.771880 
.772168 
.772457 
.772745 

.ol 

4.81 
4.80 
4.80 
4.80 
4.80 

.228408 
.228120 
.227832 
.227543 
.227255 

25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 

9.707606 
.707819 
.708032 
.708245 
.708458 
.708670 

3.55 
3.55 
3.54 
3.54 
3.54 

3KA 

9.934574 
.934499 
.934424 
.934349 
.934274 
.934199 

1.25 
1.25 
1.25 
1.25 
1.25 

1OK 

9.773033 
.773321 
.773608 
.773896 
.774184 
.774471 

4.80 
4.80 
4.80 
4.79 
4.79 
4  79 

10.226967 
.226679 
.226392 
.226104 
.225816 
.225529 

20 
19 
18 
17 
16 
15 

46 

.708882 

.D*x 

Q  KA 

.934123 

.ZD 

1OK 

.774759 

.225241 

14 

47 

.709094 

O.O4 

.934048 

•2Sv 

.775046 

47O 

.224954 

13 

48 
49 

.709306 
.709518 

3.53 
3.53 
3.53 

.933973 
.933898 

1  .25 
1.26 
1.26 

.775333 
.775621 

.  li) 

4.79 
4.78 

.224667 
.224379 

12 
11 

50 
61 
52 
63 
54 
55 
56 

9.709730 
.709941 
.710153 
.710364 
.710575 
.710786 
.710997 

3.53 
3.52 
3.52 
3.52 
3.52 
3.51 

3K1 

9.933822 
.933747 
.933671 
.933596 
.933520 
.933445 
.933369 

1.26 
1.26 
1.26 
1.26 
1.26 
1.26 

10ft 

9.775908 
.776195 
.776482 
.776769 
.777055 
.777342 
.777628 

4.78 
4.78 

4.78 
4.78 
4.78 
4.78 

477 

10.224092 
.223805 
.223518 
.223231 
.222945 
.222658 
.222372 

10 
9 
8 
7 
6 
5 
4 

57 
58 
59 
60 

.711208 
.711419 
.711629 
.711839 

.Ol 

3.51 
3.51 
3.51 

.933293 
.933217 
.933141 
.933066 

.  £\) 

1.26 
1.26 
1.26 

.777915 

.778201 

.778488 
.778774 

•  44 

4.77 
'4.77 

4.77 

.222085 
.221799 
.221512 
.221226 

3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Bine. 

D.I". 

Cotang. 

D.  1". 

Tang. 

1S7 

120° 


TABLE  IV.   LOGARITHMIC  SINES,  ETC       71 

3r                                         148° 

M.   Sine. 

D.l  . 

Cosine. 

D.l" 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 
3 
4 
5 

9.711839 
.712050 
.712260 
.712469 
.712679 
.712889 

3.50 
3.50 
3.50 
3.50 
3.49 

9.933066 
.932990 
.932914 
.932838 
.932762 
.932685 

1.27 
1.27 
1.27 
1.27 
1.27 

9.778774 
.779060 
.779346 
.779632 
.779918 
.780203 

4.77 
4.77 
4.77 
4.76 
4  76 

10.221226 
.220940 
.220654 
.220368 
.220082 
.219797 

60 

59 
68 
57 
56 
65 

6 
7 
£ 
9 

.713098 
.713308 
.713517 
.713726 

3.49 
3.49 
3.49 
3.48 
3.48 

932609 
.932533 
.932457 
.932380 

1  .27 
1.27 
1.27 
1.27 
1.27 

.780489 
.780775 
.781060 
.781346 

4.76 
4  76 
4.76 
4.76 
4.76 

.219511 
.219225 
.218940 
.218654 

64 
63 
62 
51 

10 
11 

12 

9.713935 
.714144 
.714352 

3.48 
3.48 

9.932304 
.932228 
.932151 

1.27 
1.27 

9.781631 
.781916 
.782201 

4.75 
4.75 

4TK 

10.218369 
.2*6084 
.217799 

60 
49 
48 

13 
14 
15 

.714561 
.714769 
.714978 

3.48 
3.47 
3.47 

3  AT 

.932075 
.931998 
.931921 

1.28 
1.28 
1.28 

1QQ 

.782486 
.782771 
.783056 

.75 
4.75 
4.75 

4fK 

.217514 
.217229 
.216944 

47 
46 
45 

16 

.715186 

.4if 

SAT 

.931845 

•  Zo 

100 

.783341 

•  iD 

4TK. 

.216659 

44 

17 

.715394 

.47 

.931768 

.£6 

.783626 

.  IO 

4*TA 

.216374 

43 

18 
19 

.715602 
.715809 

3.46 
3.46 
3.46 

.931691 
.931614 

1.28 
1.28 
1.28 

.783910 
.784195 

.74 
4.74 
4.74 

.216090 
.215805 

42 
41 

20 

9.716017 

9.931537 

1  28 

9.784479 

474. 

10.215521 

40 

21 

22 

.716224 
.716432 

3.'46 

.9314GO 
.931383 

l!28 

1f)0 

.784764 
.785048 

•  1  4 

4.74 

.215236 
.214952 

39 
38 

23 
24 
25 

.716639 
.716846 
.717053 

3.45 
3.45 
3.45 

.931306 
.931229 
.931152 

.ZO 

1.28 
1.29 

1   '  '  ' 

.785332 
.785616 
.785900 

4.74 
4.74 
4.73 

.214668 
.214384 
.214100 

37 
36 
35 

26 
27 
28 
29 

.717259 
.717466 
.717673 
.717879 

3.45 
3.44 
3.44 
3.44 
3.44 

.931075 
.930998 
.930921 
.930843 

1-29 
1.29 
1.-29 
1.29 

.786184 
.786468 
.786752 
.787036 

4-73 
4.73 
4.73 
4.73 
4.73 

.213816 
.213532 
.213248 
.212964 

34 
33 
32 
31 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

9.718085 
.718291 
.718497 
.718703 
.718909 
.719114 
.719320 
.719525 
.719730 
.719935 

3.43 
3.43 
3.43 
3.43 
3.43 
3.42 
3.42 
3.42 
3.42 
3.41 

9.930766 

.930688 
.930611 
.930533 
.930456 
.930378 
.930300 
.930223 
.930145 
.930067 

1.29 

1.29 

l!29 
1.29 
1.29 
1.30 
1.30 
1.30 
1.30 

9.787319 
.787603 
.787886 
.788170 
.788453 
.788736 
.789019 
.789302 
.789585 
.789868 

4.73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.71 
4.71 

0.212681 
.212397 
.212114 
.211830 
.211547 
.211264 
.210981 
.210698 
.210415 
.210132 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 

9.720140 
720345 
.720549 
.720754 
.720958 
.721162 

3.41 
3.41 
3.41 
3.41 
3.40 

9.929989 
.929911 
.929833 
.929755 
.929677 
.929599 

1.30 
1.30 
1.30 
1.30 
1.30 

19A 

9.790151 

.790433 
.790716 
.790999 
.791281 
.791563 

4.71 
4.71 
4.71 
4.71 
4.71 

4TA 

0.209849 

.209567 
.209284 
.209001 
.208719 

.208437 

20 
19 
18 
17 
16 
15 

46 

47 
48 

.721366 
.721570 
.721774 

3^40 
3.40 

0  QO 

.929521 
.929442 
.929364 

.  oU 
1.30 
1.31 

191 

.791846 
.792128 
.792410 

.70 
4.70 
4.70 

.208154 
.207872 
.207590 

14 
13 
12 

49 

.721978 

o.oy 
3.39 

.929286 

.ol 
1.31 

.792692 

4.70 
4.70 

.207308 

11 

50 

9.722181 

0   QO 

9.929207 

1«Vf 

9.792974 

4TA 

0.207026 

10 

51 
52 
53 

.722385 

.722588 
.722791 

o.ov 
3.39 
3.39 

0  OQ 

.929129 
.929060 
.928972 

.ol 
1.31 
1.31 

Iof 

.793256 
.793538 
.793819 

:.7O 

4.70 
4.70 

.206744 
.206462 
.206181 

9 
8 
7 

54 
55 

.722994 
.723197 

o.oo 
3.38 

0  00 

.928893 
.928815 

.ol 
1.31 

101 

.794101 
.794383 

4.69 
4.69 

.205899 
.205617 

6 
5 

56 
67 
58 
59 
60 

.723400 
.723603 
.723805 
.724007 
.724210 

O.Oo 

3.38 
3.37 
3.37 
3.37 

.928736 
.928657 
.928578 
.928499 
.928420 

.ol 
1.31 
1.31 
1.31 
1.32 

.794664 
.794945 
.795227 
.795508 
.795789 

4.69 
4.69 
4.69 
4.69 
4.69 

.205336 
.205055 
.204773 
.204492 
.204211 

4 
3 
2 
1 
0 

M. 

Cosine. 

D.l"  '  Sine. 

D.l". 

Cotang.  D.l". 

Tang. 

M. 

121 


68° 


72      TABLE  IV.   LOGARITHMIC  SINES,  ETC. 

32°                                            147o 

M. 

Sine. 

D.l". 

Cosine. 

D.I" 

Tang. 

D.I  . 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.724210 
.724412 
.724614 
.724816 
.725017 
.725219 
.725420 
.725622 
.725823 
.726024 

3.37 
3.37 
3.36 
3.36 
3.36 
3.36 
3.36 
3.35 
3.35 
3.35 

9.928420 
.928342 
.928263 
.928183 
.928104 
.928025 
.927946 
.927867 
.927787 
.927708 

1.32 

1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 

9.795789 
.796070 
.796351 
.796632 
.796913 
.797194 
.797475 
.797755 
.798036 
.798316 

4.68 
4.68 
4.68 
4.68. 
4.68 
4.68 
4.68 
4.68 
4.67 
4.67 

10.204211 
.203930 
.203649 
.203368 
.203087 
.202806 
.202525 
.202245 
.201964 
.201684 

60 
69 
58 
57 
66 
65 
64 
63 
62 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.726225 
.72W26 
.726626 
.726827 
.727027 
.727228 
.727428 
.727628 
.727828 
.728027 

3.35 
3.34 
3.34 
3.34 
3.34 
3.34 
3.33 
3.33 
3.33 
3.33 

9.927629 
.927549 
.927470 
.927390 
.927310 
.927231 
.927151 
.927071 
.926991 
.926911 

1.32 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

9.798596 
.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.800557 
.800836 
.801116 

4.67 
4.67 
4.67 
4.67 
4.67 
4.66 
4.66 
4.66 
4.66 
4.66 

10.201404 
.201123 
.200843 
.200563 
.200283 
.200003 
.199723 
.199443 
.199164 
.198884 

50 
49 
-48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.728227 
.728427 
.728626 
.728825 
.729024 
.729223 
.729422 
.729621 
.729820 
.730018 

3.33 
3.32 
3.32 
3.32 
3.32 
3.31 
3.31 
3.31 
3.31 
3.31 

9.926831 
.926751 
.926671 
.926591 
.926511 
.926431 
.926351 
.926270 
.926190 
.926110 

1.33 
1.33 
1.33 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 

9.801396 
.801675 
.801955 
.802234 
.802513 
.802792 
.803072 
.803351 
.803630 
.803908 

4.66 
4.66 
4.66 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
4  65 

10.198604 
.198325 
.198045 
.197766 
.197-187 
.197208 
.196928 
.196640 
.196370 
.196092 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.730216 
.730415 
.730613 
.730811 
.731009 
.731206 
.731404 
.731602 
.731799 
.731996 

3.30 
3.30 
3.30 
3.30 
3.30 
3.29 
3.29 
3.29 
3.29 
3.28 

9.926029 
.925949 
.925868 
.925788 
.925707 
.925626 
.925545 
.925465 
.925384 
.925303 

1.34 

1.34 
1.34 
1.34 
1.35 
1.35 
1.35 
1.35 
1.35 
1.35 

9.804187 
.804466 
.804745 
.805023 
.805302 
.805580 
.805859 
.806137 
.806415 
.806693 

4.65 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4  63 

10  195813 
.  195534 
.  195255 
.194977 
.194698 
.194420 
.194141 
.193863 
.193585 
.193307 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

9.732193 
.732390 
.732587 
.732784 
.732980 
.733177 
.733373 
.733569 
.733765 
.733961 

3.28 
3.28 
3.28 
3.28 
3.27 
3.27 
3.27 
3.27 
3.27 
3.26 

9.925222 
.925141 
.925060 
.924979 
.924897 
.924816 
.924735 
.924654 
.924572 
.924491 

1.35 
1.35 
1.35 
1.35 
1.35 
1.35 
1.36 
1.36 
1.36 
1.36 

9.806971 
.807249 
.807527 
.807805 
.808083 
.808361 
.808638 
808916 
.809193 
.809471 

4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.62 
4.62 
4.62 

10.193029 
.192751 
.192473 
.192195 
.191917 
.191639 
.191362 
.191084 
.190807 
.190529 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
61 
62 
53 
54 
55 
56 
67 
58 

9.734157 
.734353 
.734549 
.734744 
.734939 
.735135 
.735330 
.735525 
.735719 

3,26 
3.26 
3.26 
3.26 
3.25 
3.25 
3.25 
3.25 

9.924409 
.924328 
.924246 
.924164 
.924083 
.924001 
.923919 
.923837 
.923755 

1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.37 

9.809748 
.810025 
.810302 
.810580 
.810857 
.811134 
.811410 
.811687 
.811964 

4.62 
4.62 
4.62 
4.62 
4.62 
4.61 
4.61 
4.61 

10.190252 
.189975 
.189698 
.189420 
.189143 
.188866 
.188590 
.188313 
188036 

10 
9 
8 
7 
6 
5 
4 
3 
2 

69 

60 

.735914 
.736109 

3.24 

.923673 
.923591 

1.37 
1.37 

.812241 
.812517 

4.61 
4.61 

.187759 
.187483 

1 

0 

M 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M 

122° 


57C 


TABLE   IV.      LOGARITHMIC   SINES,   ETC.              73 

33o                                                                                                                146° 

M. 

Sine.      D.I". 

Cosine.  JD.l" 

Tang       D.I".  ! 

Cotang. 

M. 

0 

9.736109 
.736303 

3.24 

o    04 

9.923591 
.923509 

1.37 
1  37 

9.812517 
.812794 

4.C1 
4.61 

10.187483 

.187206 

60 
59 

2 

4 
5 
6 
7 
8 
9 

.736498 
.736692 
.736886 
.737080 
.737274 
.737467 
.737661 
.737855 

O./-X 

3.24 
3.23 
3.23 
3.23 
3.23 
3.23 
3.22 
3.22 

.923427 
.923345 
.923263 
.923181 
.923098 
.923016 
.922933 
.922851 

1^37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.38 

.813070 
.813347 
.813623 
.813899 
.814176 
.814452 
.814728 
.815004 

4'.61 
4.61 
4.60 
4.60 
4.60 
4.60 
4.60 
4.60 

.186930 
.186653 
.186377 
.186101 
.185824 
.185548 
.185272 
.184996 

58 
57 
56 
55 
54 
53 
52 
51 

10 

9.738048 

3  ^     9.922768 

1OQ 

9.815280 

10.184720 

50 

11 
12 
13 
14 
15 
16 
17 

.738241 
.738434 
.738627 

.73882" 
.739013 
.739206 
.739398 

3.22 
3.22  i 
3.21 
3.21 
3.21 
3.21 

.922686 

.922003 
.922520 

!  9223^5 
.922272 
.922189 

.OO 

1.38 
1.38 
1.38 
1.38 
1.38 
1.38 

.815555 
.815831 
.816107 
.816382 
.816658 
.816933 
.817209 

4'.60 
4.59 
4.59 
4.59 
4.59 
4.59 

.184445 
.184169 
.183893 
.183618 
.183342 
.183067 
.182791 

49 
48 
47 
46 
45 
44 
43 

18 
19 

.739590 
.739783 

3.21 
3.20 
3.20 

.922106 
.922023 

1.38 
1.38 
1.38 

.817484 
.817759 

4.59 
4.59 
4.59 

.182516 
.182241 

42 
41 

20 

21 
22 
23 
21 
25 
26 

9.739975 
.740167 
.740359 
.740550 
.740742 
.740934 
.741125 

3.20 
3.20 
3.20 
3.19 
3.19 
3.19 

9.921940 
.921857 
.921774 
.921691 
.921607 
.921524 
.921441 

1.39 
1.39 
1.39 
1.39 
1.39 
1.39 

9.818035 
.818310 
.818585 
.818860 
.819135 
.819410 
.819684 

4.59 
4.58 
4.58 
4.58 
4.58 
4.58 

4KQ 

10.181965 
.181690 
.181415 
.181140 

.180865 
.180590 
.180316 

40 
39 
38 
37 
36 
35 
34 

'27 

.741316 

3.19 
3m 

.921357 

1.39 

.819959 

.00 
4KQ 

.180041 

33 

28 

.741508 

.  1U 

.921274 

Ion 

.820234 

.Oo 
4    to 

.179766 

32 

29 

.741699 

3.  18 
3.18 

.921190 

.  oy 
1.39 

.820508 

.Do 
4.58 

.179492 

31 

30 
31 
32 
33 
34 
35 
36 
37 

9.741889 
.742080 
.742271 
.742462 
.742652 
.742842 
.743033 
.743223 

3.18 
3.13 
3.18 
3.17 
3.17 
3.17 
3.17 

9.921107 
.921023 
.920909 
.920856 
.920772 
JB6B8 
.920604 
.920520 

1.39 
1.39 
1.40 
1.40 
1.40 
1.40 
1.40 

9.820783 
.821057 
.821332 
.821606 
.821880 
.822154 
.822420 
.82270:; 

4.57 
4.57 
4.57 
4.57 
4.57 
4.57 
4.57 

4KT 

10.179217 
.178943 
.178668 
.178394 
.178120 
.177846 
.177571 
.177297 

30 

29 
28 
27 
26 
25 
24 
23 

38 
39 

.743413 
.743602 

3J6 
3.16 

.986488 

.920352 

l!40 
1.40 

.822977 
.823250 

.Dl 

4.57 
4.56 

.177023 
.176750 

22 
21 

40 
41 

9.743792 
.743982 

3.16 

9.920268 
..920184 

1.40 

1JA 

9.823524 
.823798 

4.56 

10.176476 
.176202 

20 

19 

42 
43 
44 
45 
46 

.744171 
.744361 
.744550 
.744739 
.744928 

3.16 
3.16 
3.15 
3.15 
3.15 

3   IK. 

.920099 
.920015 
.919931 
.919846 
.919762 

.40 
1.40 
1.41 
1.41 
1.41 
1  41 

.824072 
.824345 
.824619 
.824893 
.825166 

4.56 
4.56 
4.56 
4.56 
,    4.56 

.175928 
.175655 
.175381 
.175107 
.174834 

18 
17' 
16 
15 
14 

47 
48 
49 

.745117 
.745306 
.745494 

.  ID 

3.15 
3.14 
3.14 

.919677 
.919593 
.919508 

l!4l 
1.41 
1.41 

.825439 
.825713 
.825986 

4^56 
4.55 
4.55 

.174561 
.174287 
.174014 

13 
12 
11 

60 
51 

9.745683 
.745871 

3.14 

9.919424 
.919339 

1.41 

9.826259 
.826532 

4.55 

10.173741 
.173468 

10 
9 

62 
63 
64 
65 
66 
67 
58 
69 
60 

.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 
.747562 

3.  14 
3.14 
3.13 
3.13 
3.13 
3.13 
3.13 
3.12 
3.12 

.919254 
.919169 
.919085 
.919000 
.918915 
.918830 
.918745 
.918659 
.918574 

1  .41 
1.41 
1.41 
1.42 
1.42 
1.42 
1.42 
1.42 
1.42 

.826805 
.827078 
.827351 
.827624 
.827897 
.828170 
.828442 
.828715 
.828987 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.54 
4.54 
4.54 

.173195 
.172922 
.172649 
.172376 
.172103 
.171830 
.171558 
.171285 
.171013 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I". 

Sine. 

D.I 

Cotang. 

D.I". 

Tang. 

M. 

74     TABLE  IV.   LOGAEITHMIO  SINES,  ETC. 

84'                                             145" 

M. 

Sine. 

0.1". 

Cosine. 

D.l  . 

Tang. 

D.I  "... 

Cotang. 

M. 

0 

9.747562 

3  1O 

9.918574 

1  49 

9.828987 

4KA 

10.171013 

60 

1 

2 
3 
4 
5 

.747749 
.747936 
.748123 
.748310 

.748497 

.  14 

3.12 
3.12 
3.11 
3.11 

3-t  1 

.918489 
.918404 
.918318 
.918233 
.918147 

J  .  4.4 
1  42 
1.42 
1.42 
1.42 

1.4O 

.829260 
.829532 
.829805 
.830077 
.830349 

.0'* 

4.54 

4.54 
4.54 
4.54 

4CJ 

.170740 
.170468 
.170195 
.169923 
.169651 

59 

58 
57 
56 
55 

6 

7 
8 
9 

.748683 
.748870 
.749050 
.749243 

.  11 

3.11 
3.11 
3.10 
3.10 

.918062 
.917976 
.917891 
.917805 

.  4t> 

1.43 
1.43 
1.43 
1.43 

.830621 
.830893 
.831165 
.831437 

.  D^ 

4.53 
4.53 
4.53 
4.53 

.169379 
.  169107 
.168835 
.168563 

54 
53 
52 
51 

10 
11 
12 
13 
14 

9.749429 
.749615 
.749801 
.749987 
.750172 

3.10 
3.10 
3.10 
3.10 
3  no 

9.917719 
.917634 
.917548 
.917462 
.917376 

1.43 
1.43 
1.43 
1.43 

14  Q 

9.831709 
.831981 
x  .832253 
.832525 
.832796 

4.53 
4.53 
4.53 
4.53 

4KQ 

10.168291 
.168019 
.167747 
.167475 
.167204 

50 
49 
48 
47 
46 

15 

.750358 

.uy 

3  An 

.917290 

.43 

.830068 

.OO 

.166932 

45 

16 

.750543 

.uy 

3AQ 

.917204 

1  .43 

1/lQ 

.833339 

4.53 

4rO 

.166661 

44 

17 

18 
19 

.750729 
.750914 
.751099 

.Uif 

3.09 
3.09 
3.08 

.917118 
.917032 
.916946 

.4o 

1.44 
1.41 
1.44 

.833611 
.833882 
.834154 

Di« 

4.52 
4.52 
4.52 

.166389 
.166118 
.165846 

43 
42 
41 

20 

9.751284 

3  08 

9.916859 

1  A/1 

9.834425 

4Kf\ 

10.165575 

40 

21 
22 

.751469 
.751654 

3^08 

3f\Q 

.916773 
.916687 

J  .  44 

1.44 

.834696 
.834967 

.  &A 

4.52 

.165304 
.165033 

39 
38 

23 

.751839 

.  Oo 

3f\Q 

.916600 

1  .44 

.835238 

4.52 

.164762 

37 

24 
25 

.752023 
.752208 

.(JO 

3.07 
3  07 

.916514 
.916427 

1.44 

1.44 

1A4 

.835509 
.835780 

4.52 
4.52 

4KO 

.164491 
.164220 

£6 

35 

26 

27 

.752392 
.752576 

3^07 

3A7 

.916341 
.916254 

.  44 

1  44 

.836051 
.836322 

.&£ 

4.51 

.163949 
.163678 

34 
33 

28 
29 

.752760 
.752944 

.Ui 

3.07 
3.06 

.916167 
.916081 

1.44 
1.45 
1.45 

.836593 
.836864 

4.51 
4.51 
4.51 

.163407 
.163136 

32 
31 

30 
31 
32 
33 
34 
35 

9.753128 
.753312 
.753495 
.753679 
.753862 
.754046 

3.06 
3.06 
3.06 
3.06 
3.05 
3  Arc 

9.915994 
.915907 
.915820 
.915733 
.915646 
.9155.59 

1.45 
1.45 
1.45 
1.45 
1.45 

9.837134 

.837405 
.837675 
.837946 
.838216 

.838487 

4.51 
4.51 
4.51 
4.51 
4.51 

10.162866 
.162595 
.162325 
.162054 
.161784 
.161513 

30 

29 
28 
27 
26 
25 

36 
37 
38 
39 

.754229 
.754412 
.754595 

.754778 

.UD 

3.05 
3.05 
3.05 
3.05 

.915472 
.915385 
.915297 
.915210 

1  .45 
1.45 
1.45 
1.45 
1.46 

.838757 
.839027 
.839297 
.839568 

4.51 
4.50 
4.50 
4.50 
4.50 

.161243 
.160973 
.  160703 
.160432 

24 
23 
22 
21 

40 
41 
42 
43 

9.754960 
.755143 
.755326 

.755508 

3.04 
3.04 
3.04 

3  A/I 

9.915123 
.915035 
.914948 
.914860 

1.46 
1.46 
1.46 

9.839838 
.840108 
.840378 
.840648 

4.50 
4.50 
4.50 

10.160162 
.159892 
.159622 
.159352 

20 
19 
18 
17 

44 
45 
46 

47 
48 
49 

.755690 
.755872 
.756054 
.756236 
.756418 
.756600 

.U4 
3.04 
3.03 
3.03 
3.03 
3.03 
3.03 

.914773 
.914685 
.914598 
.914510 
.914422 
.914334 

1  .46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 

.840917 
.841187 
.841457 
.841727 
.841996 
.842260 

4.50 
4.50 
4.49 
4.49 
4.49 
4.49 
4.49 

.159083 
.158813 
.158543 
.158273 
.158004 
.157734 

16 
15 
14 
13 
12 
11 

50 
51 
52 
63 
64 
65 
66 
67 
68 
59 
60 

9.756782 
.756963 
.757144 
.757326 
.757507 
.757688 
.757869 
.758050 
.758230 
.758411 
.758591 

3.02 
3.02 
3.02 
3.02 
3.02 
3.02 
3.01 
3.01 
3.01 
3.01 

9.914246 
.914158 
.914070 
.913982 
.913894 
.913806 
.913718 
.913630 
.913541 
.913453 
.913365 

1.47 
1.47 
1.47 
1-47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 

9.842535 
.842805 
.843074 
.843343 
.843612 
.843882 
.844151 
.844420 
.844689 
.844958 
.845227 

4.49 

4.49 
4.49 
4.49 
4.49 
4.49 
4.48 
4.48 
4.48 
4.48 

10.157465 
.157195 
.156926 
.156657 
.156388 
.156118 
.155849 
.155580 
.155311 
.155042 
.154773 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I  . 

Sine. 

D.I" 

Cotang. 

D.  1  . 

Tang. 

M. 

124C 


55C 


TABLE    IV.      LOGARITHMIC   SINES,    ETC.             75 
35°                                                                                                            144° 

M. 

Sine. 

D.I   . 

Cosine. 

D.I 

Tang. 

D.I  . 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 
6 

9.758591 

.758772 
.758952 
.759132 
.759312 
.759492 
.759672 

3.01 
3.00 
3.00 
3.00 
3.00 
3.00 

2    Oft 

9.913365 
.913276 
.913187 
.913099 
.913010 
.912922 
.912833 

1.47 

1.48 
1.48 
1.48 
1.48 
1.48 

1AQ 

9.845227 
.845496 
.845764 
.846033 
.846302 
.846570 
.846839 

4.48 
4.48 
4.48 
4.48 
4.48 
4.48 

4AQ 

10.154773 
.154504 
.154236 
.153967 
.153698 
.153430 
.153161 

60 
69 

58 
67 
66 
65 

54 

7 
8 

.759852 
.760031 

.yy 
2.99 

2   GO 

.912744 
.912655 

.4o 

1.48 

1AQ 

.847108 
.847376 

.4o 

4.47 

447 

.152892 
.152624 

63 
62 

9 

.760211 

.yy 
2.99 

.912566 

.4o 

1.48 

.847644 

.41 

4.47 

.152356 

51 

10 

9.760390 

9.912477 

140 

9.847913 

4*1 

10.152087 

60 

11 

12 
13 

.760569 
.760748 
.760927 

2!99 
2.98 

.912388 
.912299 
.912210 

.4o 

1.48 
1.49 

11'  1 

.848181 
.848449 
.848717 

JBI 

4.47 
4.47 

44.7 

.151819 
.151551 
.151283 

49 
48 
47 

14 

15 
16 

.761106 
.761285 
.761464 

2!98 
2.98 

2   Oft 

.912121 
.912031 
.911942 

.4y 
1.49 
1.49 

It.. 

.848986 
.849254 
.849522 

.44 

4.47 
4.47 

4     AT 

.151014 
.150746 
.150478 

46 
45 
44 

17 

.761642 

.  yo 

rt     Q-T 

.911853 

.  .4y 

.849790 

•  4i 

4.  4fi 

.150210 

43 

18 
19 

.761821 
.761999 

2i97 
2.97 

.911763 
.911674 

1^49 
1.49 

.850057 
.850325 

4^46 
4.46 

.149943 
.149675 

42 
41 

20 
21 
22 

9.762177 
.762356 
.762534 

2.97 
2.97 
2  97 

9.911584 
.911495 
.911405 

1.49 
1.49 

9.850593 
.850861 
.851129 

4.46 
4.46 
4  46 

10.149407 
.149139 
.148871 

40 
39 
38 

23 

.762712 

2'ofi 

.911315 

1Kf\ 

.851396 

4  46 

.148604 

37 

24 
25 

.762889 
.763067 

.yo 
2.96 
2  96 

.911226 
.911136 

.ou 
1.50 

1CA 

.851664 
.851931 

4^46 
4  46 

.148336 
.148069 

36 
35 

26 
27 

28 

.763245 
.763422 
.763600 

£96 
2.96 

2QK 

.911046 
.910956 
.910866 

.OU 

1.50 
1.50 

1CA 

.852199 
.852466 
.852733 

4^46 
4.46 
44fi 

.147801 
.147534 
.147267 

34 
33 
32 

29 

.763777 

.yo 
2.95 

.910776 

.OU 

1.50 

.853001 

.40 

4.45 

.146999 

31 

30 
31 
32 
33 
34 
35 
36 

9.763954 
.764131 

.764308 
.764485 
.764662 
.764838 
.765015 

2.95 
2.95 
2.95 
2.95 
2.94 
2.94 
2  94 

9.910686 
.910596 
.910506 
.910415 
.910325 
.910235 
.910144 

1.50 
1.50 
1.50 
1.51 
1.51 
1.51 

Ir-i 

9.853268 
.853535 
.853802 
.854069 
.854336 
.854603 
.854870 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 

4AK 

10.146732 
.146465 
.146198 
.145931 
.145664 
.145397 
.145180 

30 
29 
28 
27 
26 
25 
24 

37 
38 
39 

.765191 
.765367 
.765544 

2i94 
2.94 
2.93 

.910054 
.909963 
.909873 

.01 

1.51 
1.51 
1.51 

.855137 
.855404 
.855671 

•  4t> 

4.45 
4.45 
4.44 

.144863 
.144596 
.144329 

23 

22 
21 

40 

9.765720 

200 

9.909782 

1      11 

9.855938 

4  44 

10.144062 

20 

41 

.765896 

.yo 

.909691 

1  .01 

Ie.1 

.856204 

4AA 

.143796 

19 

42 
43 

.766072 
.766247 

2i93 

.909601 
.909510 

•  Ol 

1.51 

1K1 

.856471 
.856737 

.44 

4.44 

4    A  A 

.143529 
.143263 

18 

17 

44 
45 
46 
47 

48 
49 

.766423 
.766598 
.766774 
.76C949 
.767124 
.767300 

2i93 
2.92 
2.92 
2.92 
2.92 
2.92 

.909419 
.909328 
.909237 
.909146 
.909055 
.908964 

•  01 

1.52 
1.52 
1.52 
1.52 
1.52 
1.52 

.857004 
.857270 
,857537 
.857803 
.858069 
.858336 

.44 

4.44 
4.44 
4.44 
4.44 
4.44 
4.44 

.142996 
.142730 
.142463 
.142197 
.141931 
.141664 

16 
15 
14 
13 
12 
11 

60 
51 
62 
63 

9.767475 
.767649 
.767824 
.767999 

2.91 
2.91 
2.91 

9.908873 
.908781 
.908690 
.908599 

1.52 
1.52 
1.52 

Ieo 

9.858602 

.858868 
.859134 
.R59400 

4.44 
4.43 
4.43 

10.141398 
.141132 
.140866 
.140600 

10 
9 
8 

7 

54 

.768173 

2.91 
201 

.908507 

.OJ 
Iff) 

.859666 

4.43 

4      JO 

.140334 

6 

65 
66 

f>7 

.768348 
.768522 

7/>o/»Q7 

.yi 
2.91 
2.90 

.908416 
.908324 

OAQOQQ 

•flfli 

1.53 
1.53 

.859932 
.8H0198 

ftttTUtfl 

iff 

4.43 
4.43 

.140068 
.139802 

5 

4 

Of 

68 

.  *uooy  < 
.768871 

2.90 

.yuo^oo 
.908141 

1.53 

.  OOU4U4 

.800730 

4.43 

.139536 
.139270 

3 
2 

59 
60 

.769045 
.769219 

2!90 

.908049 
.907958 

1  .53 
1.53 

.860995 
.861261 

4.43 
4.43 

.139005 
.138739 

1 
0 

"57 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

"UTi 

125' 


76     TABLE  IV.   LOGARITHMIC  SINES,  ETC. 

36°                                           143° 

M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 
1 

2 
3 

9.769219 
.769393 
.769566 
.769740 

2.90 
2.90 
2.89 

9.907958 
.907866 
.907774 
.907682 

1.53 
1.53 
1.53 

9.861261 
.861527 
.861792 
.862058 

4.43 
4.43 

4.43 

10.138739 
.138473 
.138208 
.137942 

60 

59 
68 
57 

4 
5 
6 

7 
8 

.769913 
.770087 
.770260 
.770433 

.770606 

2.89 
'2.89 
2.89 
2.89 
2.88 
200 

.907590 
.907498 
.907406 
.907314 
.907222 

1.53 
1.53 
1.53 
1.54 
1.54 

.862323 
.862589 
.862854 
.863119 
.863385 

4^42 
4.42 
4.42 

4.42 
4jif\ 

.137677 
.137411 
.137146 
.136881 
.136615 

66 
55 
54 
63 
62 

9 

.770779 

.00 
2.88 

.907129 

1  .54 
1.54 

.863650 

.4J 

4.42 

.J36350 

61 

10 

9.770952 

200 

9.907037 

9.863915 

4  Act 

10.  136085 

60 

11 

.771125 

.00 
2QQ 

.906945 

1.54 

1KX 

.864180 

.4J 
4Acy 

.135820- 

49 

12 

.771298 

.OO 
200 

.906852 

.DC: 

.864445 

'..T.A 

.135555 

48 

13 

.771470 

.00 

207 

.906760 

1  .54: 
1  F\*. 

.864710 

4  JO 

.135290 

47 

14 
15 
16 
17 
18 

.771643 
.771815 
.771987 
.772159 
.772331 

.01 

2.87 
2.87 
2.87 
2.87 

2O7 

.906667 
.906575 
.906482 
.906389 
.906290 

l!54 
1.54 
1.55 
1.55 

.864975 
.865240 
.865505 
.865770 
.866035 

[««J9 

4.42 
4.41 
4.41 
4.41 

.135025 
.134760 
.134495 
.134230 
.133965 

46 
45 
44 
43 

42 

19 

.772503 

.04 

2.86 

.906204 

1  .55 
1.55 

.866300 

4.41 
4.41 

.133700 

41 

20 

9.772G75 

ft 

9.906111 

IKK 

9.866564 

4A1 

10.133436 

40 

21 

.772847 

20/5 

.906018 

.  OD 

.866829 

:.41 

A   11 

.133171 

39 

22 
23 

.773018 
.773190 

•  oO 

2.86 

2  Oft 

.905925 
.905832 

1.55 

IKK 

.867094 
.867358 

4.41 

4.41 

4A1 

.132906 
.132642 

38 
37 

24 
25 
26 
27 

28 

.773361 
.773533 
.773704 
.773875 
.774046 

.oO 

2.85 
2.85 
2.86 
2.85 

2  OK 

.905739 
.905645 
.905552 
.905459 
.905366 

.  DD 

1.55 
1.55 
1.55 
1.56 

.867623 
.867887 
.868152 
.868416 
.868680 

.41 

4.41 
4.41 
4.41 
4.41 

4  Art 

.132377 
.132113 
.131848 
.131584 
.131320 

36 
35 
34 
33 
32 

29 

.774217 

.Oc) 

2.85 

.905272 

l!56 

.868945 

:.4U 

4.40 

.131055 

31 

30 
31 

9.774388 
.774558 

2.84 

204 

9.905179 
.905085 

1.56 

IKf* 

9.869209 
.869473 

4.40 

4AC\ 

10.130791 
.130527 

30 
29 

32 
33 

.774729 
.774899 

.cr* 

2.84 

.904992 
.904898 

.  OO 

1.56 

.869737 
.870001 

»4U 

4.40 

4  Art 

.130263 
.129999 

28 
27 

34 
35 
30 
37 

.775070 
.775240 
.775410 
.775580 

2^84 
2.84 

2.83 

200 

.904804 
.904711 
.904617 
.904523 

l!56 
1.56 
1.56 

1K7 

.870265 
.870529 
.870793 
.871057 

.41) 

4.40 
4.40 
4.40 

.129735 
.129471 
.129207 
.128943 

26 
25 
24 
23. 

38 
39 

.775750 
.775920 

.00 

2.83 
2.83 

.904429 
.904335 

.  .04 

1.57 
1.57 

.871321 
.871585 

4^40 
4.40 

.128679 
.128415 

22 
21 

40 
41 
42 
43 
44 

9.776090 
.776259 
.776429 
.776598 
.776768 

2.83 
2.83 

2.82 
2.82 

2QO 

9.904241 
.904147 
.904053 
.903959 
.903864 

1.57 
1.57 
1.57 
1.57 

It-  FT 

9.871849 
.872112 
.872376 
.872640 
.872903 

4.40 
4.39 
4.39 
4.39 

10.128151 
.127888 
.127624 
.127360 
.127097 

20 
19 
18 
17 
16 

45 
46 
47 

.776937 
.777106 
.777275 

.04 

2.82 
2.82 

2Qi> 

.903770 
.903676 
.903581 

.07 

1.57 
1.57 
1  57 

.873167 
.873430 
.873694 

4^39 
4.39 

.126833 
.126570 
.126306 

15 
14 
13 

48 

.777444 

•Odd 

.903487 

.873957 

4.0*7 

.126043 

12 

49 

1 

.777613 

2.81 
2.81 

.903392 

1  .58 
1.58 

.874220 

4.39 
4.39 

.125780 

11 

50 
61 
52 

9.777781 
.777950 
.778119 

2.81 
2.81 

0  Q1 

9.903298 
.903203 
.903108 

1.58 
1.58 

1KQ 

9.874484 
.874747 
.875010 

4.39 
4.39 

10.125516 
.125253 
.124990 

10 
9 
8 

63 
54 
55 
66 

.778287 
.778455 
.778624 
.778792 

Z.ol 

2.81 
2.80 
2.80 

2Qrt 

.903014 
.902919 
.902824 
.902729 

.Oo 

1.58 
1.58 
1.58 

1KQ 

.875273 
.875537 
.875800 
.876063 

4!  39 
4.38 
4.38 

40Q 

.124727 
.124463 
.124200 
.123937 

7 
6 
5 
4 

57 

.778960 

.oU 

20/1 

.902634 

.OO 
1KQ 

.876326 

.OO 
40Q 

.123674 

3 

68 

.779128 

.OU 

2Qrt 

.902539 

.Oo 
1  .  £9 

.876589 

.OO 
40Q 

.123411 

2 

59 

.779295 

.oU 

.902444 

.876852 

•  OO 

40Q 

.123148 

1 

60 

.779463 

2.79 

.902349 

1  .  59 

.877114 

•  OO 

.122886 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotaiig. 

D.l". 

Tang. 

M 

37' 


TABLE  IV.       LOGABITHMIC  SINES,  ETC.  77 

142° 


M. 

Sine. 

D.l" 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 
1 

9.779463 
.779631 

2.79 

9.902349 
.902253 

1.59 

9.877114 
.877377 

4.38 

10.122886 
.122623 

60 
69 

2 

.779798 

.902158 

.877640 

.122360 

r>8 

3 
4 
5 
6 

7 
8 
9 

.779966 
.780133 
.780300 
.780467 
.780634 
.780801 
.780968 

2.79 
2.79 
2.78 
2.78 
2.78 
2.78 
2.78 

.902063 
.901967 
.901872 
.901776 
.901681 
.901585 
.901490 

1.59 
1.59 
1.59 
1.59 
1.59 
1.59 
1.60 

.877903 
.878165 
.878428 
.878691 
.878953 
.879216 
.879478 

4.38 
4.38 
4.38 
4.38 
4.38 
4.37 
4.37 

.122097 
.121835 
.121572 
.121309 
.121047 
.120784 
.120522 

67 
C6 
65 
54 
63 
62 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.781134 
.781301 
.781468 
.781634 
.781800 
.781966 
.782132 
.782298 
.782464 
.782630 

2.78 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.76 
2.76 
2.76 

9.901394 
.901298 
.901202 
.901106 
.901010 
.900914 
.900818 
.900722 
.900626 
.900529 

1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.61 

9.879741 

.880003 
.880265 
.880528 
.880790 
.881052 
.881314 
.881577 
.881839 
.882101 

4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 

10.120259 
.119997 
.119735 
.119472 
.119210 
.118948 
.118686 
.118423 
.118161 
.117899 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 

22 
23 
24 
25 
26 
27 
28 
29 

9.782796 
.782961 
.783127 
.783292 
.783458 
.783623 
.783788 
.783953 
.784118 
.784282 

2.76 
2.76 
2.76 
2.75 
2.75 
2.75 
2.75 
2.75 
2.75 
2.74 

9.900433 
.900337 
.900240 
.900144 
.900047 
.899951 
.899854 
.899757 
.899660 
.899564 

1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.62 

9.882363 
.882625 
.882887 
.883148 
.883410 
.883672 
.883934 
.884196 
.884457 
.884719 

4.37 
4.37 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 

10.117637 
.117375 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.115281 

40 
39 

38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.784447 
.784612 
.784776 
.784941 
.785105 
.785269 
.785433 
.785597 
.785761 
.785925 

2.74 
2.74 
2.74 
2.74 
2.74 
2.73 
2.73 
2.73 
2.73 
2.73 

9.899467 
.899370 
.899273 
.899176 
.899078 
.898981 
.898884 
.898787 
.898689 
.898592 

1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 

9.884980 
.885242 
.885504 
.885765 
.886026 
.886288 
.886549 
.886811 
.887072 
.887333 

4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.35 
4.35 
4.35 

10.115020 
.114758 
.114496 
.114235 
.113974 
.113712 
.113451 
.113189 
.112928 
.112667 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.786089 
.786252 
.786416 
.786579 
.786742 
.786906 
.787069 
.787232 
.787395 
.787557 

2.73 
2.73 
2.72 
2.72 
2.72 
2.72 
2.72 
2.72 
2.71 
2.71 

9.898494 
.898397 
.898299 
.898202 
.898104 
.898006 
.897908 
.897810 
.897712 
.897614 

1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.C3 

9.887594 
.887855 
.888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 

4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 

10.112406 
.112145 
.111884 
.111622 
.111361 
.111100 
.110839 
.110579 
.110318 
.110057 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
55 
66 
67 
68 
69 
60 

9.787720 

.787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
.789018 
.789180 
.789342 

2.71 
2.71 
2.71 
2.71 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 

9.897516 
.897418 
.897320 
.897222 
.8*7123 
.897025 
.896926 
.896828 
.896729 
.896631 
.896532 

1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 

9.890204 
.890465 
.890725 
.890986 
.891247 
.891507 
.891768 
.892028 
.892289 
.892549 
.892810 

4.35 
4.35 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 

10.109796 
.109535 
.109275 
.109014 
.108753 
.108493 
.108232 
.107972 
.107711 
.107451 
.107190 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 

127° 


78 


TABLE   IV.      LOGARITHMIC  SINES,   ETC. 


141* 


Sine. 

D.l  . 

osine. 

D.I  . 

Tang. 

D.l". 

otang. 

M. 

0 

1 

.789342 
.789504 

2.69 
2.69 

.896532 
.896433 

1.65 
1.C5 

.892810 
.893070 

4.3i 
4.34 

.107190 
.106930 
106669 

60 
59 

KQ 

2 
3 
4 
5 
6 
7 
8 
9 

.789665 
.789827 
.789988 
.790149 
.790310 
.790471 
.790632 
.790793 

2.69 
2.69 
2.69 
2.69 
2  68 
2.68 
2.68 
2  68 

.896236 
.896137 
.896038 
.895939 
.895840 
.895741 
.895641 

1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 

.893591 
.893851 
.894111 
.894372 
.894632 
.894892 
.895152 

4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.33 
4.33 

.106409 
.106149 
.105889 
.105628 
.105368 
.105108 
.104848 

57 
56 
55 
54 
53 
52 
51 

10 
11 
32 

.790954 
.791115 
.791275 

2.68 
2.68 

9.895542 
.895443 
.895343 

1.66 
1.66 

9.895412 
.895672 
.895932 

4.33 
4.33 
4  33 

.104588 
.104329 
.104068 

50 
49 

48 

13 
14 
15 
16 
17 
18 
19 

.791436 
.791596 
.791757 
.791917 
.792077 
.792237 
.792397 

2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.66 

.895244 
.895145 
.895045 
.894945 
.894846 
.894746 
.894646 

.bo 
1.6$ 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 

.896192 
.896452 
.896712 
.896971 
.897231 
.897491 
.897751 

4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 

.103808 
.103548 
.103288 
.103029 
.102769 
.102509 
.102249 

47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.792557 
.792716 
.792876 
.793035 
.793195 
.793354 
.793514 
.793673 
.793832 
.793991 

2.66 
2.66 
2.66 
2.66 
2.66 
2.65 
2.  65 
2.65 
2.65 
2.65 

9.894546 
.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 
.893745 
.893645 

1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 

9.898010 
.898270 
.898530 
.898789 
.899049 
.899308 
.899568 
.899827 
.900086 
.900346 

4.33 
4.33 
4.33 
4.33 
4.33 
4.32 
4.32 
4.32 
4.32 
4.32 

0.101990 
.101730 
.101470 
.101211 
.100951 
.100692 
.100432 
.100173 
.099914 
.099654 

40 

39 
38 

36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.794150 
.794308 
.794467 
.794626 
.794784 
.794942 
.795101 
.795259 
.795417 
.795575 

2.65 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.63 
2.63 

9.893544 
.893444 
.893343 
.893243 
.893142 
.893041 
.892940 
.892839 
.892739 
.892638 

1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 

9.900605 
.900864 
.901124 
.901383 
.901642 
.901901 
.902160 
.902420 
.902679 
.902938 

4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 

0.099395 
.099136 
.098876 
.098617 
.098358 
.098099 
.097840 
.097580 
.097321 
.097062 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 

4: 

9.795733 
.795891 
.796049 
.796206 
.796364 
.796521 

2.63 
2.63 
2.63 
2.63 
2.62 

9  892536 
.892435 
.892334 
.892233 
.892132 
.892030 

1.69 
1.69 
1.69 
1.69 
1.69 

9.903197 
.903456 
.903714 
.903973 
.904232 
.904491 

4.32 
4.32 
4.31 
4.31 
4.31 

10.096803 
.096544 
.096286 
.096027 
.095768 
.095509 

20 
19 
18 
17 
16 
15 

( 

.796679 
.796836 
.796993 
.797150 

2.62 
2.62 
2.62 
2.62 
2.61 

.891929 
.891827 
.891726 
.891624 

1.69 
1.69 
1.69 
1.6 

.904750 
.905008 
.905267 
.905526 

4.31 
4.31 
4.3 
4.3 

.095250 
.094992 
.094733 
.094474 

14 
13 
12 
11 

50 

9.797307 

9.891523 

9.90578 

LO.  094215 

10 

5 

.797464 

.891421 

.90604 

.093957 

9 

52 
5 
54 
5 
56 
5 
5 
5 

.797621 
.797777 
.797934 
.798091 
.798247 
.798403 
.798560 
.798716 

2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.60 

.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 
.890605 

1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 

.90630 
.90656 
.90681 
.90707 
.90733 
.90759 
.90785 
.90811 

4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 

.093698 
.093440 
.093181 
.092923 
.092664 
.092406 
.092148 
.091889 

8 
7 
6 
5 
.4 
3 
2 
1 

60 

.798872 

2.60 

.890503 

1.70 

.90836 

.091631 

0 

M 

Cosine. 

D.r 

Sine. 

D.r 

.  Cotang 

D.l".  |  Tang. 

M. 

TABLE  IY.      LOGABITHMIO  SINES,  ETa  79 

140 


M.      Sine.      D.I".     Cosine.  D.I'.     Tang.      D.I".    Cotang. 


>.  798872 


.799184 


.799651 


.800117 
.800272 

9.800427 
.800582 
.800737 


.801047 
.801201 
.801356 
.801511 
.801665 
.801819 

9.801973 
.802128 
.802282 
.802436 


.802743 


.803050 
.803204 


>.803511 
.803664 
.803817 


.804123 
.804276 
.804428 
.804581 
.804734 


.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 
.806254 


9.806557 


.807011 
,807163 
.807314 
.807465 
.807615 
.807766 
.807917 
.808067 


2.60 
2.60 
2.60 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 

2.58 
2.58 
2.58 
2.58 
2.58 
2.58 
2.57 
2.57 
2.57 
2.57 

2.57 
2.57 
2.57 
2.56 
2.56 
2.56 
2.56 
2.56 
2.56 
2.55 

2.55 
2.55 
2.55 
2.55 
2.55 
2.55 
2.54 
2.51 
2.54 
2.54 

2.54 
2,54 
2.54 
2.53 
2.53 
2.53 
2.53 
2.53 
2.53 
2.52 

2.52 
2.52 
2.52 
2.52 
2.52 
2.52 
2.51 
2.51 
2.51 
2.51 


9.890503 
.890400 
.890298 
.890195 


9.889477 
.889374 
.889271 


.888755 
.888651 
.888548 

9.888444 


.887926 
.887822 
.887718 
.887614 
.887510 

K887406 
.887302 
.887198 
.887093 


.886571 
.8864G6 


.885837 


.885627 
.885522 


9.885311 
.885205 
.885100 


.884783 
.884677 
.884572 


1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 

1.72 
1.72 
1  72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 

1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.74 

1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1  74 
1.75 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.76 

1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.77 
1.71 


.909144 
.909402 


.910177 
.910435 


9.910951 
.911209 
.911467 
.911724 
.911982 
.912240 


.912756 
.913014 
.913271 


.913787 
.914044 
.914302 
.914560 
.914817 
.915075 
.915332 
.915590 
.915847 

.916104 
.916362 
.916619 
.916877 
.917131 
.917391 
.91764S 
.917900 


.918420 

9.918677 
.918934 
.919191 
.919443 
.919705 
.919962 
.920219 
.920476 
.920733 


9.921247 

.921503 
.921760 
.922017 
.922274 
.922530 
.922787 


4  30 
4  30 
4.30 
4.iJO 
4  30 
4.30 
4.30 
4.30 
4.30 
4.30 

4.30 

4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 

4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 

4.28 

4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 

4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 


0091631 
.091372 
.091114 

.090856 
.090598 
.090340 
.090082 
.089823 
.089565 
.089307 

0.089049 

.088791 
.088533 


.087760 
.087502 
.087244 
.086986 
.086729 
0086471 


086213  89 


37 


.OS54-JO 
.085183 
.084925 
.084608 
.084410 
.084153 

0.083896 

.083638 
.083381 
.083123 


.081838 
.081580 

10.081323 
.081066 
.080809 

.080552 


.079781 
.079524 
.079267 
.079010 

10.078753 
.078497 
.078240 
.077983 
.077726 
.077470 
.077213 
.076956 
.076700 
.076443 
.076187 


5O° 


Cosine. 


D.I".      Sine.      D.I"    Cotang. 


D.  1".      Tang. 


129° 


80 

40° 


TABLE   IV.      LOGARITHMIC   SINES,   ETC. 


M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 

9.808067 
.808218 
.808368 
.808519 
.808669 
.808819 

2.51 
2.51 
L.51 
2.50 
2.50 
9  50 

9.884254 
.884148 
.884042 
.883936 
.883829 
.883723 

1  77 
1.77 
1.77 
1.77 
1.77 
1  77 

9.923813 
.924070 
.924327 
.924583 
.924840 
.925096 

4.28 
4.28 
4,27 
4.27 
4.27 

4  07 

10.076187 
.075930 
.075673 
.075417 
.075160 
.074904 

60 
59 

58 
57 
56 
55 

6 

7 
8 
9 

.808969 
809119 
.809269 
.809419 

z.ou 
2.50 
2.50 
2.50 
2.50 

.883617 
.883510 
.883404 
.883297 

1  .  i  4 

1.77 
1.77 

1.78 
1.78 

.925352 
.925609 
.925865 
.926122 

:.Z< 

4.27 
4.27 
4.27 
4.27 

.074648 
.074391 
.074135 
.073878 

54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.809569 
.809718 
.809868 
.810017 
.810167 
.810316 
.810465 
.810614 
.810763 
.810912 

2.49 
2.49 
2.49 
2.49 
2.49 
2.49 
2.48 
2.48 
2.48 
2.48 

9.883191 

.883084 
.882977 
.882871 
.882764 
.882657 
.882550 
.882443 
.882336 
.882229 

1.78 
1.78 
1.78 
1.78 
1.78 
1.78 
1.78 
1.79 
1.79 
1.79 

9.926378 
.926634 
.926890 
.927147 
.927403 
.927659 
.927915 
.928171 
.928427 
.928683 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 

10.073622 
.073366. 
.073110 
.072853 
.072597 
.072341 
.072085 
.071829 
.071573 
.071317 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 

9.811061 
.811210 
.811358 
.811507 
.811655 
.811804 
.811952 
.812100 

2.48 
2.48 
2.48 
2,47 
2.47 
2.47 
2.47 

2  AT 

9.882121 
.882014 
.881907 
.881799 
.881692 
.881584 
.881477 
.881369 

1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 

1Q{\ 

9.928940 
.929196 
.929452 
.929708 
.929964 
.930220 
.930475 
.930731 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.26 

10.071060 
.070804 
.070548 
.070292 
.070036 
.069780 
.069525 
.069269 

40 
39 
38 
37 
36 
35 
34 
33 

28 

.812248 

.4< 

2AJ7 

.881261 

.oU 

.930987 

4na 

.069013 

32 

29 

.812396 

.4< 

2.47 

.881153 

1^80 

.931243 

.  Zo 
4.26 

.068757 

31 

30 

9.812544 

2A.fi 

9.881046 

9.931499 

10.068501 

30 

31 
32 

.812692 
.812840 

,4O 

2.46 

O  AjR 

.880938 
.880830 

i.'so 

.931755 
.932010 

4  '.26 

.068245 
.067990 

29 
28 

33 

.812988 

2A(* 

.880722 

1  'flA 

.932266 

4  Of? 

.067734 

27 

34 

.813135 

,4o 

.880613 

1.80 

.932522 

,OQ 

.067478 

26 

35 

.813283 

2.46 

.880505 

1QA 

.932778 

4.26 

.067222 

25 

36 
37 
38 
39 

.813430 
.813578 
.813725 
.813872 

2^46 
2.45 
2.45 
2.45 

.880397 
.880289 
.880180 
.880072 

.  oU 

1.81 
1.81 
1.81 
1.81 

.933033 
.933289 
.933545 
.933800 

4^26 
4.26 
4  26 
4.26 

.066967 
.066711 
.066455 
.066200 

24 
23 
22 
21 

40 

9.814019 

Q  JK 

9.879963 

1Q1 

9.934056 

4f>fi 

10.065944 

20 

41 
42 
43 
44 

.814166 
.814313 
.814460 
.814607 

2!45 
2.45 
2.45 
2  A^ 

.879855 
.879746 
.879637 
.879529 

.  Ol 

1.81 
1.81 
1.81 
81 

.934311 
.934567 
.934823 
.935078 

.Zo 
4.26 
4.26 
4.26 

.065689 
.065433 
.065177 
.064922 

19 
18 
17 
16 

45 

.814753 

2AA 

.879420 

.ol 
Q1 

.935333 

4  Of* 

.064667 

15 

46 

.814900 

.44 
O  A£ 

.879311 

.ol 

89 

.935589 

,Zo 

.064411 

14 

47 

48 

.815046 
.815193 

2.4A 

.879202 
.879093 

.  o  J 

.82 

.935844 
.936100 

4!  26 

.064156 
.063900 

13 
12 

49 

.815339 

2!44 

.878984 

'.82 

.936355 

4.26 
4.26 

.063645 

11 

50 
51 

9.815485 
.815631 

2.44 

9.878875 
.878766 

.82 

9.936611 
.936866 

4  26 

10.063389 
.063134 

10 
9 

52 

.815778 

2  A  ft 

.878656 

QO 

.937121 

4.Zo 

.062879 

8 

53 

.815924 

,4o 

2  A*> 

.878547 

.oZ 

00 

.937376 

4  OK 

.062624 

7 

54 

.816069 

•  4o 

.878438 

.  oZ 

.937632 

.  Zt) 
4  OK< 

.062368 

6 

55 

.816215 

2AO 

.878328 

QO 

.937887 

4  OK 

.062113 

5 

56 
57 
58 

.816361 
.816507 
.816652 

.4o 

2  A3 
2.43 

2  An 

.878219 
.878109 
.877999 

.  OO 

.83 
.83 

QO 

.938142 
.938398 
.938653 

.  ZD 

4.25 
4.25 

4  OK 

.061858 
.061602 
.061347 

4 
3 
2 

59 
60 

.816798 
.816943 

•me 

2.42 

.877890 
.877780 

.  oO 

.83 

.938908 
.939163 

.  ZD 

4.25 

.061092 
.060837 

1 
0 

IT 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

1*7. 

TABLE    IV.      LOGARITHMIC    SINES.    ETC.               81 
41°                                                                                                        138° 

M. 

Sine. 

D.i  . 

Cosine.  |  D.I". 

Tang.    |  D.I   .  j  Cotang. 

M. 

0 
1 

2 
3 

9.816943 

.817088 
.817233 
.817379 

2.42 
2.42 

2.42 

9.8777SO 
.877670 
.8775CO 
.877450 

83 
.83 

.83 

9.50163 
.939418 
.939673 
.939928 

4.25 
4.25 

4.25 

4O«t 

10.060837 
.060582 
.060327 
.060072 

60 
59 

T7 

4 

.817524         £'% 

.877340 

Ql 

.940183 

.^D 

4  °5 

.059817 

C6 

5 

.817668         o'TT 

.  cl 
C  1 

.940438 

4    OK 

.059562 

C5 

6 

.817813         «•;}        .877120 

.0* 

.940694 

.  -  D 

4  25 

.059306 

54 

7 

.817958  I      7,  ,1 

.877010 

C  1 

.940949 

4  25 

.059051 

53 

8 
9 

.818103 
.818247 

A.ti. 
2.41 
2.41 

.876899 
.876789 

.  89 

.84 
.84 

.941204 
.941458 

4^25 
4.25 

.058796 
.058542 

ro 
51 

10 

11 

9:818392 
.818536 

2.41 

2«  1 

9.876678 
.876568 

1.81 

9.941713 
.941968 

4.25 
4  25 

10.058287 
.058032 

EO 
49 

12 
13 
14 
13 

13 

.818681 
.818825 
.818969 
.819113 
.819257 

.  i  L 

2.40 
2.41 
2.40 
2.40 

.876457 
.876347 
.876236 
.876125 
.«76014 

i!si 

1.84 

1.85 

.85 

op? 

.942223 
.942478 
.942733 
.942986 
.943243 

4!25 
4.25 
4.25 
4.25 
4o*t 

.057777 
.057522 
.057267 
.057012 
.056707 

48 
47 
48 
45 
44 

17 

.819401 

o   tn 

.875904 

.CO 
OK 

.943498 

-  4-D 

4     OK 

.056502 

43- 

13 
19 

.819545 
.819689 

2^40 
2.39 

.875793 
.875682 

.CO 

.85 
.85 

.943752 
.944007 

.  -.  y 
4.25 
4.25 

.056248 
.OGC993 

42 
41 

20 
21 
22 
23 

9.819832 
.819976 
.820120 
.820263 

2.39 
2.39 
2.39 

9.875571 
.875459 
.875348 
.875237 

.S5 
85 
.85 

9.944262 
.944517 
.944771 
.945026 

4.25 
4.25 
4  24 

10.055738 
.0.35483 
.055229 
.054974 

40 

39 
38 
37 

24 
23 

.820406 
.820550 

2!.39 

.875126 
.875014 

.88 

.945281 
.945535 

4^24 

4O4 

.054719 
.054465 

36 
35 

23 

.820693 

29ft 

.874903 

« 

.945790 

:.  --t 

.054210 

34 

27 

.820836 

•  oO 

2r>0 

.874791 

.86 

'.946045 

J         « 

.053955 

33 

23 

.820979 

.00 
f>      OQ 

.874680 

.86 

Of* 

.946299 

4.24 
4  24 

.053701 

32 

29 

.821122 

2.38 

.874568 

.00 
.86 

.946554 

4'.24 

.053446 

31 

30 

9.821265 

20Q 

9.874456 

Qf* 

9.946808 

4    Of 

10.053192 

30 

31 
32 

.821407 
.821550 

.Jo 
2.38 

200 

.874344 
.874232 

.eo 
.86 

.947063 
.947318 

.  Jl 
4.24 

4OA 

.052937 
.052682 

29 
28 

33 
31 

.821693 
.821835 

.*X> 

2  37 

.874121 
.874009 

'.SI 

.947572 
.947826 

.  Z4 

4.24 

.052428 
.052174 

27 
26 

33 

O  "* 

cr 

C3 
39 

.821977 
.822120 
.822262 
.822404 
.822546 

2.37 
2.37 
2.37 
2.37 
2.37 
2.37 

.873896 
.873784 
.87:3672 
.873560 
.873448 

.87 
87 
.87 
.87 
.87 
.87 

.948081 
948336 
.948590 
.948844 
.949099 

4.24 
4.24 
4.24 
4.24 
424 
4.24 

.051919 
.051664 
.051410 
.051156 
.050901 

25 
24 
23 
22 
21 

40 
41 
42 

9.822688 
.822830 
.822972 

2.37 
2.36 

9.873335 
.873223- 
.873110 

.87 

.88 

9.949353 
.949608 
.949862 

4.24 
4.24 

10.050647 
.050392 
.050133 

20 
19 
18 

43 

.823114 

2.36 

.872998 

.88 

.950116 

}•?}  !     .049884 

17 

44 

.823253 

2.36 

.88 

oo 

.950371 

m.M 

.049629 

16 

45 

.823397 

2w 

's72772 

.00 

.950625 

4.24 

j     Of 

.049375 

15 

46 
47 

.823539 
.823680 

.00 
2.36 

.872659 
.872547 

.88 
.88 

950879 
.951133 

4!24 

4O4 

.049121 

.048867 

14 
13 

48 
49 

.823821 
.823963 

2.  35 
2.35 

.872434 
.872321 

!88 
1.88 

.951388 
.951642 

»2M 

4.24 
4.24 

.048612 
.048358 

12 
11 

50 
51 

9.824104 
.824243 

2.35 

9.872208 
.872095 

.89 

9.951896 
.932150 

4.24 

10.048104 
.047850 

10 
9 

52 
53 
54 

55 

.824386 
.824527 
.824668 
.824808 

2.35 
2.35 
2.35 
2.35 

204 

.871981 
.871868 
.871755 
.871641 

.89 
1.89 
1.89 
1.89 

.952405 
.952659 
.952913 
.953167 

4.24 
4.24 
4.24 
4.24 

4rtJ 

.047595 
.047341 
.047087 
.046833 

8 
7 
6 
5 

56 
57 
58 

J) 

.824949 
.825090 
.825230 
.825371 
.825511 

.  vr* 

2.34 
2.34 
2.34 
2.34 

.871528 
.871414 
.871301 
.871187 
.871073 

1.89 
1.89 
1.89 
1.89 
1.90 

.953421 
.953675 
.953929 
.954183 
.954437 

.Ji 

4.24 
4.23 
4.23 
4.23 

.046579 
.046325 
.046071 
.045817 
.045563 

4 
3 

2 

0 

M. 

Cosine. 

D,l  .  i     Sine. 

Dl    . 

Cotaner. 

1   D.I   . 

Tang.      M. 

"53 


48* 


TABLE  IV.   LOGARITHMIC  SINES,  ETC. 
42°                                              137^ 

M. 

Sine. 

D.I  . 

Cosmo. 

D.I  . 

Tang. 

D.I". 

Cotung. 

M. 

0 

1 
2 
3 
4 
5 

9.825511 
.825651 
.825791 
.825931 
.826071 
.826211 

2.34 
2.34 
2.33 
2.33 
2.33 

9.871073 
.870960 
.870846 
.870732 
.870618 
.870504 

1.90 
.90 
.90 
.90 
.90 

Ofi 

9.954437 
.954691 
.954945 
.955200 
.955454 
.955707 

4.23 
4.23 
4.23 
4.23 
4.23 

10.045563 
.045309 
.045055 
.044800 
.044546 
.044293 

60 
59 
58 
57 
56 
55 

6 

7 
8 

.826351 
.826491 
.826631 

2  '.33 
2.33 

.870390 
.870276 
.870161 

.  yu 
.90 
.90 

Ol 

.955961 
.956215 
.956469 

4i23 
4.23 

.044039 
.043785 
.043531 

54 
53 
52 

9 

.826770 

2  '.33 

.870047 

.yi 
.91 

.956723 

4.23 
4.23 

043277 

51 

10 

9.826910 

200 

9.869933 

Q1 

9.956977 

400 

10.043023 

50 

11 
12 

.827049 

.827189 

.  oZ 

2.32 
2  32 

.869818 
.869704 

.yi 
.91 

Ol 

.957231 
.957485 

•  Zo 
4.23 

400 

.042769^ 
.042515 

49 
48 

13 
14 
15 
16 

17 
18 
19 

.827328 
.827467 
.827606 
.827745 
.827884 
.828023 
.828162 

2^32 
2.32 
2.32 
2.32 
2.31 
2.31 
2.31 

.869589 
.869474 
.869360 
.869245 
.869130 
.869015 
.868900 

.  yi 
.91 
.91 
.91 
.91 
.92 
.92 
.92 

.957739 
.957993 
.958246 
.958500 
.958754 
.959008 
.959262 

.  Zo 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

.042261 
.042007 
.041754 
.041500 
.041246 
.040992 
.040738 

47 
46 
45 
44 
43 
42 
41 

20 
21 
22 

9,828301 
.828439 

.828578 

2.31 

2.31 

2O1 

9.868785 
.868670 
.868555 

.92 
.92 

OO 

9.959516 
.959769 
.960023 

4.23 
4.23 

400 

10.040484 
.040231 
.039977 

40 
39 
38 

23 

.828716 

.ol 

20.1 

.868440 

.  .  '  '  -_ 

.960277 

.  Zo 

.039723 

37 

24 
23 

.828855 
.828993 

.ol 

2.31 

2  on 

.868324 
.868209 

!92 
oo 

.960530 
.960784 

4.23 
4.23 

4"  OQ 

.039470 
.039216 

36 
35 

26 
27 
28 

.829131 
.829269 
.829407 

.oU 

2.30 
2.30 

0  OA 

.868093 
.867978 
.867862 

.  yz 
.93 
.93 

.961038 
.961292 
.961545 

.2.6 
4.23 
4.23 
4"  oo 

.038962 
.038708 
.038455 

34 
33 
32 

29 

.829545 

Z*o(J 
2.30 

.867747 

'93 

.961799 

.2,6 
4.23 

.038201 

31 

30 

9.829683 

2  on 

9.867631 

9.962052 

400 

10.037948 

30 

31 

32 

.829821 
.829959 

.oU 
2.30 

2  on 

.867515 
.867399 

!93 

.962306 
.962560 

.  Zo 

4.23 

400 

.037694 
.037440 

29 

28 

33 
34 
35 

.830097 
.830234 
.830372 

.ZJ 

2.29 
2.29 
200 

.867283 
.867167 
.867051 

!93 
.93 

.962813 
.963067 
.963320 

•  Zo 

4.23 
4.23 

400 

.037187 
.036933 
.036680 

27 
26 
25 

36 
37 

.830509 
.830646 

.  ZJ 
2.29 

.866935 
.866819 

!94 

.963574 
.963828 

tZo 

4.23 

.036426 
.036172 

24 
23 

38 
39 

.830784 
.830921 

2.29 
2.29 
2.29 

.866703 
.866586 

.94 
.94 
.94 

.964081 
.964335 

4.23 
4.23 
4.23 

.035919 
.035665 

22 
21 

40 

9.831058 

20Q 

9.866470 

9.964588 

4  OO 

10.035412 

20 

41 
42 

.831195 
.831332 

•  Zo 

2.28 

.866353 
.866237 

!94 

.964842 
.965095 

.  zz 
4.22 

.035158 
.034905 

19 
18 

43 
44 

.831469 
.831606 

2.28 
2.28 

.866120 
.866004 

.94 
.94 

.965349 
.965602 

4.22 
4.22 

.034651 
.034398 

17 
16 

45 

.831742 

2.28 

.865887 

.95 

.965855 

4.22 

.034145 

15 

46 
47 

48 

.831879 
.832015 
.832152 

2.  28 

2.28 
2.27 

.865770 
.865653 
.865536 

.95 
.95 
.95 

.966109 
.966362 
.966616 

4.22 
4.22 
4.22 

.033891 
.033638 
.033384 

14 
13 
12 

49 

.832288 

2.27 
2.27 

.865419 

.95 
.95 

.966869 

4.22 

4.22 

.033131 

11 

50 

9.832425 

9.865302 

9.967123 

10.032877 

10 

51 

52 
53 

.832561 
.832697 
.832833 

2.27 
2.27 

2.27 

.865185 
.865068 
.864950 

.95 
.95 
.95 

.967376 
.967629 
.967883 

4.22 
4.22 
4.22 

.032624 
.032371 
.032117 

9 

8 
7 

54 
55 
56 

.832969 
.833105 
.833241 

2.27 
2.27 
2.26 

.864833 
.864716 
.864598 

.96 
.96 
.96 

.968136 
.968389 
.968643 

4.22 
4.22 
4.22 

.031864 
.031611 
.031357 

6 
5 
4 

57 
58 
59 

.833377 
.833512 
.833648 

2.26 
2.26 
2.26 

.864481 
.864363 
.864245 

.96 
.96 
.96 

.968896 
.969149 
.969403 

4.22 
4.22 
4.22 

.031104 
.030851 
.030597 

3 
2 

1 

60 

.833783 

2.26 

.864127 

.96 

.969656 

4.22 

.030344 

0 

M. 

Cosine. 

Dr. 

Sine. 

D.r  . 

Cotang. 

D.r  . 

Tang. 

M. 

132° 


4T 


43° 


TABLE   IV.      LOGAKITHMIG   SINES,    ETC.  83 

136* 


M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Ootang. 

M. 

0 

9.833783 

• 

9.864127 

9.969656 

4  22 

10.030344 

60 

1 
2 

3 
4 
5 
6 

.833919 
.834054 
.834189 
.834325 
.834400 
.834595 

2^26 
2.25 
2.25 
2.25 
2.25 

2  OK 

.864010 

.863892 
.863774 
.863656 
.863538 
.863419 

1^97 
.97 
.97 
.97 
.97 

.969909 
.970162 
.970416 
.970669 
.970922 
.9711-75 

4^22 
4.22 
4.22 
4.22 
4.22 

A   .,.  » 

.030091 
.029838 
.029584 
.029331 
.029078 
.028825 

59 
58 
57 
66 
55 
54 

7 
8 
9 

.834730 
•8348G5 
.834999 

.  JO 

2.25 
2.25 
2.25 

.863301 
.863183 
.863064 

.97 
.97 
.97 

.971429 
.971682 
.971935 

4i22 
4.22 
4.22 

.028571 
.028318 
.028065 

53 
52 
51 

ib 
11 

3.835134 
.835269 

2.24 

9.862946 
.862827 

.98 

9.972188 
.972441 

4.22 

10.027812 
.027559 

60 

49 

12 

.835403 

2.24 

.862709 

.98 

.972694 

4.22 

.027306 

48 

13 

.835538 

2.24 

.862590 

.98 

flQ 

.972948 

4.22 
400 

.027052 

47 

14 

.835672 

2.24 

.862471 

.98 

.973201 

lot 

.026799 

46 

15 

.835807 

2.24 

.862353 

.98 

.973454 

4.22 

.026546 

45 

16 

.835941 

2.24 

.862234 

.98 

.973707 

4.22 
400 

.026293 

44 

17 

.836075 

2.24 

.862115 

.98 

.973960 

.at 

.02C040 

43 

18 

.830209 

2.23 

.861996 

.98 

.974213 

4.22 

.025787 

42 

19 

.836343 

2.23 
2.23 

.861877 

.98 
.99 

974466 

4.22 
4.22 

.025534 

41 

20 

9.836477 

9.861758 

9.974720 

10.025280 

40 

21 

.836611 

2.23 

.861638 

.99 

.974973 

4.22 

.025027 

39 

22 

.836745 

2.23 

.861519 

.99 

.975226 

4.22 

.024774 

S3 

23 

.836878 

2.23 

.861400 

.99 

.975479 

4.22 

.024521 

37 

24 

.837012 

2.23 

.861280 

.99 

.975732 

4.22 

.02-1268 

S6 

25 

.837146 

2.23 

.861161 

.99 

.975985 

4.22 

.024015 

35 

26 

.837279 

2.22 

.861041 

.99 

.976238 

4.22 

.02S762 

34 

27 
28 
29 

.837412 
.837546 
.837679 

2.22 
2.22 
2.22 
2.22 

.860922 
.860802 
.860682 

.99 
2.00 
2.00 
2.00 

.976491 
.976744 

.976997 

4.22 
4.22 
4.22 
4.22 

.022509 
.02C256 
.02C003 

33 

C2 
SI 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.837812 
.837945 
.838078 
.838211 
.838344 
.838477 
.838610 
.838742 
.838875 
.839007 

2.22 
2.22 
2.22 
2.21 
2.21 
2.21 
2.21 
'  2.21 
2.21 
2.21 

9.860562 
.860442 
.860322 
.860202 
.860082 
.859962 
.859842 
.859721 
.859601 
.869480 

2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.01 
2.01 
2.01 
2.01 

">  "77250 
.^7503 
.977756 
.978009 
.978262 
.978515 
.978768 
.979021 
.979274 
.979527 

4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

10.022750 
.022497 
.022244 
.021991 
.021738 
.021485 
.021232 
.020979 
.020726 
.020473 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 

9.839140 

A  *%1 

9.859360 

9.979780 

10.020220 

20 

41 
42 
43 
44 
45 
46 
47 
43 
49 

.839272 
.839404 
.839536 
.839668 
.839800 
.839932 
.840064 
.846196 
.840328 

2.21 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.19 
2.19 

.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.858393 
.858272 

2.01 
2.01 
2.01 
2.01 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 

.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.981550 
.981803 
.982056 

4.22 
4.22 
4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4  21 
4.21 

.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
62 
63 
54 
55 
56 
67 
58 
59 
CO 

9.840459 
.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841378 
.841509 
.841640 
.841771 

2.19 
2.19 
2.19 
2.19 
2.19 
2.19 
2.18 
2.18 
2.18 
2.18 

9.858151 
.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
.857056 
.856934 

2.02 
2.02 
2.02 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 

9.982309 
.982562 
.982814 
.983067 
.983320 
.983573 
.983826 
984079 
.984331 
.984584 
.984837 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

10.017691 
.017438 
.017186 
.016933 
.016680 
.016427 
.016174 
.015921 
.015669 
.015416 
.015163 

10 
9 

7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I  . 

Sine. 

D.I" 

rotanc. 

D.  1  . 

Tane. 

M. 

46° 


84      TABLE  IV.   LOGARITHMIC  SINES,  ETC. 

44°                                              135° 

M. 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

M. 

0 
1 
2 
3 
4 

9.841771 
.841902 
.842033 
.842163 
.842294 

2.18 
2.18 
2.18 
2.18 

21  7 

9.856934 
.856812 
.856690 
.856568 
.856446 

2.03 
2.04 
2.04 
2.04 
2ni 

9.984837 
.985090 
.985343 
.985596 

.985848 

4.21 
4.21 
4.21 
4.21 

10.015163 
.014910 
.014657 
.014404 
.014152 

60 
59 
58 
57 
56 

5 

.842424 

.  14 

217 

.856323 

,\r± 

.986101 

4O1 

.013899 

55 

6 

.842555 

•11 

.856201 

O   i 

.986354 

.  Zl. 
4  O1 

.013646 

54 

7 

.842685 

2.  17 

21  7 

.856078 

2f\A 

.986607 

,zi 

401 

.013393 

53 

8 
9 

.842815 
.842946 

.  14 

2.17 
2.17 

.855956 
.855833 

.  U-i 

2.04 
2.04 

.986860 
.987112 

.  Zl 

4.21 
4.21 

.013140 
.012888 

52 
51 

10 
11 

9.843076 
.843206 

2.17 

O  17 

9.855711 
.855588 

2.05 

9.987365 
.987618 

4.21 

4  O1 

10.012635 
.012382 

50 
49 

12 
13 
14 
15 
16 
17 

.843336 
.843466 
.843595 
.843725 
.843855 
.843984 

Z.  14 

2.16 
2.16 
2.16 
2.16 
2.16 

21  £S 

.855465 
.855342 
.855219 
.855096 
.854973 
.854850 

2^05 
2.05 
2.05 
2.05 
2.05 

.987871 
.988123 
.988376 
.988629 
.988882 
.989134 

.  ZL 

4.21 
4.21 
4.21 
4.21 
4.21 

4  O1 

.012129 
.011877 
.011624 
.011371 
.011118 
.010866 

48 
47 
46 
45 
44 
43 

18 
19 

.844114 
.844243 

lb 
2.16 
2.16 

.854727 
.854603 

2.05 
2.06 
2.06 

.989387 
.989640 

•  Zi 
4.21 
4.21 

.010613 
.010360 

42 
41 

20 
21 

22 

9.844372 
.844502 
.844631 

2-15 
2.15 

9.854480 
.854356 
.854233 

2.06 

2.06 

9.989893 
.990145 
.990398 

4.21 
4.21 

4O1 

10.010107 

.009855 
.009602 

40 

39 
38 

23 
24 
25 
26 
27 
28 
29 

.844760 
.844889 
.845018 
.845147 
.845276 
.845405 
.845533 

2.  15 
2.15 
2.15 
2.15 
2.15 
-  2.15 
2.14 
2.14 

.854109 
.853986 
.853862 
.853738 
.853614 
.853490 
.853366 

2.06 
2.06 
2.06 
2.06 
2.06 
2.07 
2.07 
2.07 

.990651 
.990903 
.991156 
.991409 
.991662 
.991914 
.992167 

.ZL. 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

.009349 
.009097 
.008844 
.008591 
.008338 
.008086 
.007(333 

37 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.845662 
.845790 
.845919 
.846047 
.846175 
.846304 
.846432 

2.14 
2.14 
2.14 
2.14 
2.14 
2.14 
210 

9.853242 
.853118 
.852994 
.852869 
.852745 
.852620 
.852496 

2.07 

2.07 
2.07 
2.07 
2.07 
2.08 
2  no 

9.992420 
.992672 
.992925 
.993178 
.993430 
.993683 
.993936 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4  O1 

10.007580 
.007328 
.007075 
.006822 
.006570 
.006317 
.006064 

30 
29 
28 
27 
26 
25 
24 

37 
38 
39 

.846560 
.846688 
.846816 

.lo 
2.13 
2.13 
2.13 

.852371 
.852247 
.852122 

.Uo 
2.08 
2.08 
2.08 

.994189 
.994441 
.994694 

./I 

4.21 
421 

4.21 

.005811 
.005559 
.005306 

23 
22 
21 

40 

9.846944 

210 

9.851997 

2f\Q 

9.994947 

4  O1 

10.005053 

20 

41 

.847071 

.  lo 
2  10 

.851872 

.  Uo 

2f\Q 

.995199 

•  /I 
4  O1 

.004801 

19 

42 
43 
44 
45 
46 
47 
48 
49 

.847199 
.847327 
.847454 
.847582 
.847709 
.847836 
.847964 
.848091 

.lo 
2.13 
2.13 
2.12 
2.12 
2.12 
2.12 
2.12 
2.12 

.851747 
.851622 
.851497 
.851372 
.851246 
.851121 
.850996 
.850870 

.Us 
2.08 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 

.995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.996968 
.997221 

.Zl 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

.004548 
.004295 
.004043 
.003790 
.003537 
.003285 
.003032 
.002779 

18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
62 
63 
54 
55 
56 
57 
58 
59 

9.848218 
.848345 
.848472 
.848599 
.848726 
.848852 
.848979 
.849106 
.849232 
.849359 

2.12 
2.12 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 

9.850745 
.850619 
.850493 
.850368 
.850242 
.850116 
.849990 
.849864 
.849738 
.849611 

2.09 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 

9.997473 
.997726 
.997979 
.998231 
.998484 
.998737 
.998989 
.999242 
.999495 
.999748 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

10.002527 
.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 
.000505 
.000252 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

60 

.849485 

2.11 

.849485 

2.11 

10.000000 

4.21 

.000000 

0 

M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 

45° 


TABLE  V. 


LATITUDES   AND   DEPARTURES, 

OR 

TRAVERSE   TABLE. 


86         TABLE  V.   TBAVERSE  TABLE* 

B'ng 

Di*t.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4.   Dist.  5. 

B'ng 

0    , 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat  |  Dep. 

Lat. 

Dep. 

•  / 

0  15 

1.0000 

0.0044 

2.0000 

0.0087 

3.0000 

0.01314.00000.0175 

5.0000 

0.0218 

89  45 

30 

0000 

0087 

1.9999 

0175 

2.9999 

0262 

3.9998 

0349-4.9998 

0436 

30 

45 

0.9999 

0131 

9998 

0262 

9997 

0393 

9997 

0524 

9996 

0654 

15 

1  0 

9998 

0175 

9997 

0349 

9995 

0524 

9994 

0698 

9992 

0873 

89  0 

15 

9998 

0218 

9995 

0436 

9993 

0654 

9990 

0873 

9988 

1091 

45 

30 

9997 

0262 

9993 

0524 

9990 

0785 

9986 

1047 

9983 

1309 

30 

45 

9995 

0305 

9991 

0611 

9986 

0916 

9981 

1222 

9977 

1527 

15 

2  0 

9994 

0349 

9988 

0698 

9982 

1047 

9976 

1396 

9970 

1745 

88  0 

15 

9992 

0393 

9985 

0785 

9977 

1178 

9969 

1570 

9961 

1963 

45 

30 

9990 

0436 

9981 

0872 

9971 

1309 

9962 

1745 

9952 

2181 

30 

45 

0.9988 

0.0480 

1.9977 

0.0960 

2.9965 

0.1439 

3.9954 

0.1919 

4.9942 

0.2399 

15 

3  0 

9986 

0523 

9973 

1047 

9959 

1570 

9945 

2093 

9931 

2617 

87  0 

15 

9984 

0567 

9968 

1134 

9952 

1701 

993f 

2268 

9920 

2835 

45 

30 

9981 

0610 

9963 

1221 

9944 

1831 

9925 

2442 

9907 

3052 

30 

45 

9979 

0654 

9957 

1308 

9936 

1962 

9914 

2616 

9893 

3270 

15 

4  0 

9976 

0698 

9951 

1395 

9927 

2093 

9903 

2790 

9878 

3488 

86  0 

15 

9973 

0741 

9945 

1482 

9918 

2223 

9890 

2964 

9863 

3705 

45 

30 

9969 

0785 

9938 

1569 

9908 

2354 

9877 

3138 

9846 

3923 

30 

45 

9966 

0828 

9931 

1656 

9897 

2484 

9863 

3312 

9828 

4140 

15 

5  0 

9962 

0872 

9924 

1743 

9886 

2615 

9848 

3486 

9810 

4358 

85  0 

15 

0.9958 

0.0915 

1.9916 

0.1830 

2.9874 

0.2745 

3.9832 

0.3660 

4.9790 

0.4575 

45 

30 

9954 

0958 

9908 

IP1.? 

9862 

2875 

9816 

3834 

9770 

4792 

30 

45 

9950 

1002 

9899 

20  '* 

9849 

3006 

9799 

4008 

9748 

5009 

15 

6  0 

9945 

1045 

9890 

2091 

9836 

3136 

9781 

4181 

9726 

5226 

84  0 

15 

9941 

1089 

9881 

2177 

9822 

3266 

9762 

4355 

9703 

5443 

45 

30 

9936 

1132 

9871 

2264 

9807 

3396 

9743 

4528 

9679 

5660 

30 

45 

9931 

1175 

9861 

2351 

9792 

3526 

9723 

4701 

9653 

5877 

15 

7  0 

9925 

1219 

9851 

2437 

9776 

3656 

9702 

4875 

9627 

6093 

83  0 

15 

9920 

1262 

9840 

2524 

9760 

3786 

9680 

5048 

9600 

6310 

45 

30 

9914 

1305 

9829 

2611 

9743 

3916 

9658 

5221 

9572 

6526 

30 

45 

0.9909 

0.1349 

1.9817 

0.2697 

2.9726 

0.4046 

3.9635 

0.5394 

4.9543 

0.6743 

15 

8  0 

9903 

1392 

9805 

2783 

9708 

4175 

9611 

5567 

9513 

6959 

82  0 

15 

9897 

1435 

9793 

2870 

9690 

4305 

9586 

5740 

9483 

7175 

45 

30 

9890 

1478 

9780 

2956 

9670 

4434 

9561 

5912 

9451 

7390 

30 

45 

9884 

1521 

9767 

3042 

9651 

4564 

9534 

6085 

9418 

7606 

15 

9  0 

9877 

1564 

9754 

3129 

9631 

4693 

9508 

6257 

9384 

7822 

81  0 

15 

9870 

1607 

9740 

3215 

9610 

4822 

9480 

6430 

9350 

8037 

45 

30 

9863 

1650 

9726 

3301 

9589 

4951 

9451 

6602 

9314 

8252 

30 

45 

9856 

1693 

9711 

3387 

9567 

5080 

9422 

6774 

9278 

8467 

15 

10  0 

9848 

1736 

9C96 

3473 

9544 

5209 

9392 

6946 

9240 

8682 

80  0 

15 

0.9840 

0.1779 

1.9681 

0.3559 

2.9521 

0.5338 

3.9362 

0.7118 

4.9202 

0.8897 

45 

30 

9833 

1822 

9665 

3645 

9498 

5467 

9330 

7289 

9163 

9112 

30 

45 

9825 

1865 

9649 

3730 

9474 

5596 

9298 

7461 

9123 

9326 

15 

11  0 

9816 

1908 

9633 

3816 

9449 

5724 

9265 

7632 

9081 

9540 

79  0 

15 

9808 

1951 

9616 

3902 

9424 

5853 

9231 

7804 

9039 

9755 

45 

30 

9799 

1994 

9598 

3987 

9398 

5981 

9197 

7975 

8996 

9968 

30 

45 

9790 

2036 

9581 

4073 

9371 

6109 

9162 

8146 

8952 

1.0182 

15 

12  0 

9781 

2079 

9563 

4158 

9344 

6237 

9126 

8316 

8907 

0396 

78  0 

15 

9772 

2122 

9545 

4244 

9317 

6365 

9089 

8487 

8862 

0609 

45 

30 

9763 

2164 

9526 

4329 

9289 

6493 

9052 

8658 

8815 

0822 

30 

45 

0.9753 

0.2207 

1.9507 

0.4414 

2.9260 

0.6621 

3.9014 

0.8828 

4.8767 

.1035 

15 

13  0 

9744 

2250 

9487 

4499 

9231 

6749 

8975 

8998 

8719 

1248 

77  0 

15 

9734 

2292 

9468 

4584 

9201 

6876 

8935 

9168 

8669 

1460 

45 

30 

9724 

2334 

9447 

4669 

9171 

7003 

8895 

9338 

8618 

1672 

30 

45 

9713 

2377 

9427 

4754 

9140 

7131 

8854 

9507 

8567 

1884 

15 

14  0 

9703 

2419 

9406 

4838 

9109 

7258 

8812 

9677 

8515 

2096 

76  0 

15 

9692 

2462 

9385 

4923 

9077 

7385 

8769 

9846 

8462 

2308 

45 

30 

9681 

2504 

9363 

5008 

9044 

7511 

8726 

1.0015 

8407 

2519 

30 

45 

9670 

2546 

9341 

5092 

9011 

7638 

8682 

0184 

8352 

2730 

15 

15  0 

9659 

2588 

9319 

5176 

8978 

7765 

8637 

0353 

8296 

2941 

75  0 

•  ' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

•  ' 

B'ng 

Dlst.  1. 

Dist.  2. 

Dist.  3. 

Dlst.  4. 

Dlst.  5. 

B'ng 

TABLE  V.      TKAVERSE  TABLE. 


Liijt.  6. 

i>ist.  7.    iMst.  8.  :  Dist.  9.    Dist.  1O. 

B'ag 

,  .  Lat.  Dep.  Lat. 

Dep, 

Lat. 

Dep. 

Lat.  J  Dep. 

Lat.  Dep. 

,  , 

0  15  5.  9999  0.  0262  6.  99990.  0305*7.  9999  0.0349  8.  9999  0.0393;9.  9999  0.0436  89  45 

30  9998  '  0524 

9997 

0611 

9997)  0698!  9997 

0785 

9996  0873   30 

45  9903  0785 

9994 

0916 

9993   1047 

9992 

1178 

9991   1309   15 

1  0  9991   1047 

9989 

1222 

9988  !  1396 

9986 

1571 

9985 

174589  0 

15  9986  1309 

9983 

1527 

9981   1745 

9979  1963 

9976 

2181  j   45 

30   9979   1571 

9976 

1832 

9973  2094 

9969 

2356 

9966 

26181   30 

45l  9972   1832 

9967 

'2138 

9963  2443 

9958 

2748 

9953 

3054!   15 

2  0  9963  2094 

9957 

2443  9951   2792  99451  3141 

9939 

3490 

88  0 

15  9l»54   2.r,»; 

9946|  2748)  9938'  3141  1  99311  3533 

9923 

3926 

45 

30  9943  2617 

9933 

3053 

9924 

3490)  9914 

3926 

9905 

4362 

30 

455.99310.28796.9919 

0.3358 

7.9908 

0.3838 

8.9896 

0.43189.9885.0.4798 

15 

3  0  9918  3140  9904 

3664 

9890 

4187 

9877 

4710J  9863  5234 

87  0 

15  9904  3402 

9887 

3968 

9871 

4535 

9855 

5102 

98391  5669 

45 

30  9888 

3663 

9869 

4273 

9851 

4884 

9832 

5494 

9813 

6105 

30 

45  9872 

3924 

9850 

4578 

9829 

5232 

9807 

5886 

9786 

6540   15 

4  o!  9854 

4185 

9829 

4883 

9805 

5581 

9781 

6278 

9756  697686  0 

15   9835 

4447 

9808  5188 

9780 

5929 

9753  6670 

9725  7411 

45 

30;  9815 

4708 

9784  5492 

9753 

6277   9723  7061   9692  7846 

30 

45 

9794'  4968 

9760 

5797 

9725 

6625  9691   7463  9657  8281 

15 

5  0 

9772  5229 

9734 

6101 

96S6 

6972   9658   7844 

9619  8716 

85  0 

15 

5.97480.5490 

6.97060.6405 

7.9664 

0.73208.96220  8235 

9.9580;0.9150 

45 

30 

9724   5751 

907*   6709 

9632 

7668 

9586'  8626 

9540  9585 

30 

45 

969* 

6011 

9648;  70131  9597 

8015 

9547 

9017 

9497:1.0019 

15 

6  0 

9671 

9617   7317 

9562 

8362 

9507 

9408 

9452  0453 

84  0 

15 

9643  6532 

9584i  7621 

9525 

8709 

9465 

9798 

9406  0887 

45 

30 

9614  6792 

9550:  79241  9486 

9056 

9421  1.0188 

9357   1320 

30 

45  9584   7052  !  9515!  8228 

9445 

9403 

9376  0578  j  9307 

17.-4 

15 

7  0  95,'3  7312!  9478|  8531 

9404 

9750 

93291  0968  9255 

218783  0 

15  9520 

7572  9440!  8834 

93601.0096 

92801  1358  9200 

2620   45 

30  9487 

7832  9401   9137 

9316!  0442 

9230  1747  9144 

3053   30 

455.9452 

0.8091  6.93(31  0.9440 

7.92691.0788 

8.91781.21379.90871.3485   15 

8  0   941H   83501  9319!  9742   9221   1134'  9124   2526!  9027   391782  0 

15   9379   8610   92761.0044   9172   1479   90C9   2914   8965   4349   45 

30 

9341 

88691  9231 

0347 

9121   1825 

9011 

3303 

89021  4781 

30 

45 

9302 

9127  9185 

0649 

9069;  2170 

8953 

3691 

8836 

5212 

15 

9  0 

9261 

9386  9138 

0950 

9015!  2515 

8892 

4079 

8769 

564381  0 

15 

9220 

9645  9090 

1252 

8960 

2859 

8830 

4467 

8700 

6074 

45 

30 

9177 

9903  9040 

1553 

8903 

3204 

8766 

4854 

8629 

6505 

30 

45 

9133 

1.0161  1  8989 

1854 

8844 

3548 

8700 

5241 

8556 

6935   15 

10  0 

9088 

0419J  8937 

2155 

8785 

3892 

8633 

5628 

8481 

736580  0 

155.9042 

1.  0677  i  6.  8883 

1.2456 

7.8723 

1.42358.8564 

1.60159.8404 

1.7794!   45 

30 

8995 

0934 

8828 

2756 

8660 

4579 

8493 

6401 

8325 

8224 

30 

45 

8947 

1191 

8772 

3057 

.8596 

4922 

8421 

6787 

8245 

8652 

15 

11  0 

8898 

1449 

8714 

3357 

85301  5265 

8346 

7173 

8163 

9081 

79  0 

15 

8847 

1705 

8655 

3656 

8463!  5607 

8271 

7558 

8079 

9509 

45 

30 

8795 

1962 

8595 

3956 

8394;  5949 

8193 

7943 

7992 

9937 

30 

45 

8743 

2219 

8533 

42-55 

8324  6291 

8114 

8328 

7905 

2.0364 

15 

12  0 

8689 

2475 

8470 

4554 

6633 

8033 

8712 

7815'  0791 

78  0 

15  8634 

2731 

8406  4852 

81781  6974,  7951 

9096 

7723 

1218 

45 

30,  8578 

2986 

8341   5151 

8104 

7315  78G7 

9480  7630 

1644 

30 

455.8521 

1.32426.8274 

1  5449 

7.8027 

1.76568.7781 

1.98639.7534 

2.2070 

15 

13  0  8462  i  3497;  8206 

5747 

7950 

7996  7693 

2.0246   7437 

249577  0 

15  8403;  3752!  8137 

6044 

7870 

8336  7604 

0628  7338 

2920 

45 

30 

8342  4007 

8066 

6341 

7790 

8676  7513 

1010  7237 

3345 

30 

45 

8281 

4261 

7994  6638 

7707 

9015  7421 

1392  7134 

3769 

15 

14  0 

8218 

4515 

79211  6935  7624 

93.34 

7327 

1773 

7030 

4192 

76  0 

15 

8154 

4769 

7846 

7231   7538  9692 

7231 

2154 

6923 

4615 

45 

30 

8089 

5023 

7770 

7527  7452  2.0030 

7133 

2534 

6815 

5038 

30 

45 

8023 

5276 

7693 

7822  7364  0368 

7034 

2914 

6705 

5460 

15 

15  0 

7956 

5529 

7615 

8117  7274 

0706 

6933 

3294 

6593 

5882 

75  0 

•  '  '  Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  j  Lat. 

Dep. 

Lat. 

.  . 

B'ngi  Dist.  6. 

Dist.  7.    Dist.  8. 

Dist.  9. 

Dist.  1O.  ;B'ng 

88         TABLE  V.   TRAVERSE  TABLE. 

B'ng 

I>ist.  1. 

Dist.  3. 

Dist.  3. 

I»is 
Lat. 

t.4. 

IMst.  5. 

B'ng 

.  . 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Dep. 

Lat. 

Dep. 

0    . 

15  15 

0.9648 

0.2630 

1.9296 

0.5261 

2.8944 

0.7891 

3.8591 

1  .0521 

4.8239 

1.3152 

7445 

30  9636 

2672 

9273 

5345 

8909 

8017 

8545 

0690 

8182 

3362 

30 

45  9625 

2714 

9249 

5429 

8874 

8143 

8498 

0858 

8123 

3572 

15 

16  0 

9613 

2756 

9225 

5513 

8838 

8269 

8450 

1025 

8063 

3782 

74  0 

15 

9600 

2798 

9201 

5597 

8801 

8395 

8402 

1193 

8002 

3991 

45 

30 

9588 

2840 

9176 

5680 

8765 

8520 

8353 

1361 

7941 

4201 

30 

45 

9576 

2882 

9151 

5764 

8727 

8646 

8303 

1528 

7879 

4410 

15 

17  0 

9563 

2924 

9126 

5847 

8689 

8771 

8252 

1695 

7815 

4619 

73  0 

15 

9550 

2965 

9100 

5931 

8651 

8896 

8201 

1862 

7751 

4827 

45 

30 

9537 

3007 

9074 

6014 

8612 

9021 

8149 

2028 

7686 

5035 

30 

45 

0.9524 

0.3049 

.9018 

0.6097 

2.8572 

0.9146 

3.8096 

1.2195 

4.7620 

.5243 

15 

18  0 

9511 

3090 

9021 

6180 

8532 

9271 

8042 

2361 

7553 

5451 

72  0 

15 

9497 

3132 

8994 

6263 

8491 

9395 

7988 

2527 

7485 

5658 

45 

30 

9483 

3173 

8966 

6346 

8450 

9519 

7933 

2692 

7416 

5865 

30 

45 

9469 

3214 

8939 

6429 

8408 

9643 

7877 

2858 

7347 

6072 

15 

19  0 

9455 

3256 

8910 

6511 

8366 

9767 

7821 

3023 

7276 

6278 

71  0 

15 

9441 

3297 

8882 

6594 

8323 

9891 

7764 

3188 

7204 

6485 

45 

30 

9426 

3338 

8,853 

6676 

8279 

.0014 

7706 

3352 

7132 

6690 

30 

45 

9412 

3379 

8824 

6758 

8235 

0138 

7647 

3517 

7059 

6896 

15 

20  0 

9397 

3420 

8794 

6840 

8191 

0261 

7588 

3681 

6985 

7101 

70  0 

15 

0.9382 

0.3461 

.8764 

0.6922 

2.8146 

.0384 

3.7528 

.3845 

4.6910 

1.7306 

45 

30 

9367 

3502 

8733 

7004 

8100 

0506 

7467 

4008 

6834 

7510 

30 

45 

9351 

3543 

8703 

7086 

8054 

0629 

7405 

4172 

6757 

7715 

15 

21  0 

9336 

3584 

8672 

7167 

8007 

0751 

7343 

4335 

6679 

7918 

69  0 

15 

9320 

3624 

8640 

7249 

7960 

0873 

7280 

4498 

6600 

8122 

45 

30 

9304 

3665 

8608 

7330 

7913 

0995 

7217 

4660 

6521 

8325 

30 

45 

9288 

3706 

8576 

7411 

7864 

1117 

7152 

4822 

6440 

8528 

15 

22  0 

9272 

3746 

8544 

7492 

7816 

1238 

7087 

4984 

6359 

8730 

68  0 

15 

9255 

3786 

8511 

7573 

7766 

1359 

7022 

5146 

6277 

8932 

45 

30 

9239 

3827 

8478 

7654 

7716 

1481 

6955 

5307 

6194 

9134 

30 

45 

0.9222 

0.3867 

.8444 

0.7734 

2.7666 

.1601 

3.6888 

1.5468 

4.6110 

1.9336 

15 

23  0 

9205 

3907 

8410 

7815 

7615 

1722 

6820 

5629 

6025 

9537 

67  0 

15 

9188 

3947 

8376 

7895 

7564 

1842 

6752 

5790 

5940 

9737 

45 

30 

9171 

3987 

8341 

7975 

7512 

1962 

6682 

5950 

5853 

9937 

30 

45 

9153 

4027 

8306 

8055 

7459 

2082 

6612 

6110 

5760 

^.0137 

15 

24  0 

9135 

4067 

8271 

8135 

7406 

2202 

6542 

6269 

5677 

0337 

66  0 

15 

9118 

4107 

8235 

8214 

7353 

2322 

6470 

6429 

5588 

0536 

45 

30 

9100 

4147 

8199 

8294 

7299 

2441 

6398 

6588 

5498 

0735 

30 

45 

9081 

4187 

8163 

8373 

7244 

2560 

6326 

6746 

5407 

0933 

15 

25  0 

9063 

4226 

8126 

8452 

7189 

2679 

6252 

6905 

5315 

1131 

65  0 

15 

0.9045 

0.4266 

1.8089 

0.8531 

2.7134 

1.2797 

3.6178 

1.7063 

4.5223 

2.1328 

45 

30 

9026 

4305 

8052 

8610 

7078 

2915 

6103 

7220 

5129 

1526 

30 

45 

9007 

4344 

8014 

8689 

7021 

3033 

6028 

7378 

5035 

1722 

15 

26  0 

8988 

4384 

7976 

8767 

6964 

3151 

5952 

7535 

4940 

1919 

64  0 

15 

8969 

4423 

7937 

8846 

6906 

3269 

5875 

7692 

4844 

2114 

45 

30 

8949 

4462 

7899 

8924 

6848 

3386 

5797 

7848 

4747 

2310 

'  30 

45 

8930 

4501 

7860 

9002 

6789 

3503 

5719 

8004 

4649 

2505 

15 

27  0 

8910 

4540 

7820 

9080 

6730 

3620 

5640 

8160 

4550 

2700 

63  0 

15 

8890 

4579 

77SO 

9157 

6671 

3736 

5561 

8315 

4451 

2894 

45 

30 

8870 

4617 

7740 

9235 

6610 

3852 

5480 

8470 

4351 

3087 

30 

45 

0.8850 

0.4656 

1.7700 

0.9312 

2.6550 

1.3968 

3.5400 

1.8625 

4.4249 

2.3281 

15 

28  0 

8829 

4695 

7659 

9389 

6488 

4084 

5318 

8779 

4147 

3474 

62  0 

15 

8809 

4733 

7618 

9466 

6427 

4200 

5236 

8933 

4045 

3666 

45 

30 

8788 

4772 

7576 

9543 

6365 

4315 

5153 

9086 

3941 

3858 

30 

45 

8767 

4810 

7535 

9620 

6302 

4430 

5069 

9240 

3836 

4049 

15 

29  0 

8746 

4848 

7492 

9696 

6239 

4544 

4985 

9392 

3731 

4240 

61  0 

15 

8725 

4886 

7450 

9772 

6175 

4659 

4900 

9545 

3625 

4431 

45 

30 

8704 

4924 

7407 

9848 

6111 

4773 

4814 

9697 

3518 

4621 

30 

45 

8682 

4962 

7364 

9924 

6046 

4886 

4728 

9849 

3410 

4811 

15 

30  0 

8660 

5000 

7321 

1.0000 

5981 

5000 

4641 

2.0000 

3301 

5000 

60  0 

•  , 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

.    0 

B'ng 

IMst.  1. 

JMst.  2. 

Dist.  3.  1  IHst.  4. 

IMst.  5. 

B'ng 

TABLE  V.   TRAVERSE  TABLE.         89 

B"'ng 

Dist.  6. 

lust.  7.    l>ist.  8.   ui»t  .  9. 

DiSt.  10. 

B'ng 

0  . 

Lafe  |  Dep. 

Lat. 

Dep. 

Lat.  JDep. 

Lat. 

Dep, 

Lat.  Dep. 

.  . 

15  15 

5.78871.57826.75351.84127.71832.10428.68312.3673 

9.6479  2.6303 

7445 

30 

7818  6034   7454!  8707   7090  1379  6727  4051 

6363 

6724 

30 

45 

7747 

62861  7372,  9001i  6996i  1715  6621  4430 

6246 

7144 

15 

16  0 

7676 

6538 

7288  9295  6901,  2051  6514  4807 

6126  i  7564 

74  0 

15 

7603 

6790 

72031  9588;  6804 

2386 

6404 

5185 

6005  7983 

45 

30 

7529 

7041 

7117  9881  1  6706 

2721 

6294 

5561 

5882 

8402 

30 

45 

7454 

7292 

7030  2.0174  6606  3056 

6181 

5938  5757 

8820 

15 

17  0 

7378 

7542 

6941;  0466  6504!  3390 

6067 

63131  5630  9237 

73  0 

15 

7301 

7792 

6851:  0758;  6402  |  3723 

5952 

6689 

5502J  9654 

45 

30 

7223  8042 

6760  1049  6297 

4056 

5835 

7064 

5372'3.0071 

30 

45 

5.7144'l.8292 

6.66682.1341 

7.61922.43898.5716 

2.7438 

9.  5240  '3.0486 

15 

18  0 

7063  8541 

6574 

1631 

60851  4721 

5595 

7812 

5106  0902 

72  0 

15 

6982 

8790 

6479 

1921 

5976  5053  5473 

8185 

4970J  1316 

45 

30 

6899 

9038 

6383 

2211 

5866  5384  5349 

8557 

4832 

1730 

30 

45 

6816 

9286 

6285 

2501 

5754  5715  !  5224 

8930 

4693 

2144 

15 

19  0 

6731 

9534 

6186 

2790  5641  i  6045J  5097 

9301 

4552 

2557 

71  0 

15 

6645 

9781 

6086 

3078 

5527  6375'  4968 

9672 

4409 

2969 

45 

30 

6558 

2.0028 

59851  3366  5411,1  6705  4838 

3.0043 

4264 

3381 

30 

45 

6471 

0275 

"5882  3654  5294  7033  4706 

0413 

4118 

3792 

15 

20  0 

6382 

0521 

5778  3941   5175 

7362i  4572 

0782 

3969 

4202 

70  0 

15 

5.6291 

2.0767 

6.5673  2.4228  7.5055  2.7689  8.4437 

3.1151 

9.3819 

3.4612 

45 

30 

6200 

1012 

55671  4515!  4934  8017  4300 

1519  3667 

5021 

30 

45 

6108 

1257 

5459J  4800  4811 

8343 

4162 

1886  3514 

5429 

15 

21  0 

6015 

1502 

5351!  5086  4686 

8669 

4022 

22531  3358 

5837 

69  0 

15 

5920 

1746 

5241 

5371!  4561   8995 

3881 

2619 

3201 

6244 

45 

30 

5825 

1990 

5129 

5655  4433  9320 

3738 

2985 

3042 

6650 

30 

45 

5729 

2233 

5017   5939;  4305!  9645 

3593 

3350 

2881 

7056 

15 

22  0 

5631 

2476 

4903  6222  4175  9969 

3447 

3715 

2718 

7461 

68  0 

15 

5532 

2719 

4788  -  6505  4043  3.0292 

3299  4078 

2554 

7865 

45 

30  5433 

2961  46721  6788;  3910j  0615 

3149 

4442 

2388 

8268 

30 

45  5.5332  2.3203  6.4554  2.7070  7.3776  3.0937 

8.29983.4804 

9.2220 

3.8671 

15 

23  0  5230  3444  4435  7351   3640  1258 

2845  5166 

2050 

9073 

67  0 

15 

5127 

3685 

4315 

76321  3503 

1580 

2691 

5527 

1879 

9474 

46 

30 

5024 

3925 

4194 

7912 

3365 

1900 

2535J  5887 

1706 

9875 

30 

45 

4919 

4165 

4072 

8192 

3225 

2220 

23781  6247 

1531 

4.0275 

15 

24  0 

4813 

4404 

3948 

8472 

3084 

2539 

2219 

6606 

1355 

0674 

66  0 

15 

4706 

4643 

3823 

8750 

2941 

2858 

2059 

6965 

1176 

1072 

45 

30 

4598 

4882 

3697 

9029 

2797 

3175 

1897 

7322 

0996 

1469 

30 

45  4489 

5120 

3570 

9306 

2651 

3493 

1733 

7679 

0814 

1866 

15 

25  0  4378 

5357 

3442 

9583 

2505 

3809 

1568 

8036 

0631 

2262 

65  0 

155.4267 

2.5594 

6.3312 

2.9860 

7.23563.41258.1401 

3.8391 

9.0446 

4.2S57 

45 

30 

4155 

5831 

3181 

3.0136 

2207 

4441   1233 

8746 

0259 

3051 

30 

45 

4042 

6067 

3049 

0411 

2056 

4756!  1063 

9100 

0070 

3445 

15 

26  0 

3928 

6302 

2916 

0686 

1904 

5070 

0891 

9453 

8.9879 

3837 

64  0 

15 
30 

3812 
3696 

6537 
6772 

2781 
•2645 

0960 
1234 

17501  5383 
1595  5696 

0719  9806 
05444.0158 

9687 
9493 

4229 
4620 

45 
30 

45 

3579 

7006 

2509 

15071  1438  6008 

0368 

0509 

9298 

5010 

15 

27  0 

3460 

7239 

2370 

17791  1281 

6319 

0191 

0859 

9101 

5399 

63  0 

15 

3341 

7472 

2231 

2051   1121 

6630 

0012 

1209 

8902 

5787 

45 

30 

3221 

7705 

2091 

2322 

0961 

6940  7.9831 

1557 

8701 

6175 

30 

45 

5.3099 

2.7937 

6.1949  3.2593  7.0799  3.  7249  17.9649  4.1905 

8.8499 

4.6561 

15 

28  0 

2977 

8168 

18061  2863  0636  7558 

9465  2252 

8295 

6947 

62  0 

15 

2853 

8399 

1662!  3132  0471!  786C 

9280;  2599 

8089 

7332 

45 

30 

2729 

8630 

1517 

3401   0305  8173 

9094 

2944 

7882 

7716 

30 

45 

2604 

8859 

1371 

3669  01381  8479 

8905 

3289 

7673 

8099 

15 

29  0 

2477 

9089 

1223 

39376.9970  8785 

8716 

3633 

7462 

8481 

61  0 

15 

2350 

9317 

1075 

4203 

9800  9090 

8525  3976 

7250 

8862 

45 

30 

2221 

9545 

0925 

4470 

9628;  9394 

8332 

4318 

7036 

9242 

30 

45 

2092 

9773 

0774 

4735 

9456:  9697 

8138 

4659 

6820 

9622 

15 

30  0 

1962  3.0000  0622 

5000 

92824.0000  7942 

5000 

6603 

5.0000 

60  0 

•  ' 

. 
Dep. 

Lat.  Dep.  Lat. 

Dep.  Lat.  Dep. 

Lat. 

Dep. 

Lat 

•  i 

B'ng 

Dist.  6. 

Blfct.  7. 

Dial.  8.  i  lH»t.9. 

J>iMt.lO.  B'ng 

G 

90 


TABLE  V.   TEA  VERSE  TABLE. 


B'ng 

J>ist.  1.  |  J*ist.  2. 

l>ist.  3. 

J>i*i.  4.   I>ist.  5. 

B'ng 

.  , 

Lat. 

Dep 

Lat. 

Dep 

Lat 

Dep. 

Lat. 

Dep 

Lat. 

Dep. 

0    , 

30  15 

0.8638 

0.503* 

1.727 

1.007 

2.591 

1.511 

3.455£ 

2.015 

4.3192 

2.518S 

5945 

30 

8616 

507£ 

7233 

015 

584 

522 

446 

0302 

308 

5377 

30 

45 

8594 

5112 

7188 

022 

578 

533 

437 

045 

2970 

5565 

15 

31  0 

8572 

51  5C 

7142 

030 

571 

545 

428 

0602 

2858 

5752 

59  0 

15 

8549 

5188 

7098 

037 

564 

5563 

419 

075 

2746 

5939 

45 

30 

8526 

5225 

7053 

0450 

557 

5675 

4106 

0900 

2632 

6125 

30 

45 

8504 

5262 

7007 

0524 

551 

5786 

4014 

1049 

2518 

6311 

15 

32  0 

8480 

5299 

6961 

0598 

544 

5898 

3922 

1197 

2402 

6496 

58  0 

15 

8457 

5336 

6915 

0672 

5372 

6003 

3829 

1345 

2286 

668 

45 

30 

8434 

5373 

6868 

0746 

5302 

6119 

3736 

1492 

2170 

6865 

30 

45 

0.8410 

0.5410 

1.6821 

1.0819 

2.5231 

1.6229 

3.3642 

2.1639 

4.2052 

2.7049 

15 

33  0 

8387 

5446 

6773 

0893 

5160 

6339 

3547 

1786 

1934 

7232 

57  0 

15 

8363 

5483 

6726 

0966 

5089 

6449 

3451 

1932 

1814 

7415 

45 

30 

8339 

5519 

6678 

1039 

5017 

6558 

3355 

2077 

1694 

7597 

30 

45 

8315 

5556 

6629 

1111 

4944 

6667 

3259 

2223 

1573 

7779 

15 

34  0 

8290 

5592 

6581 

1184 

4871 

6776 

3162 

2368 

1452 

7960 

56  0 

15 

8266 

5628 

6532 

1256 

4798 

6884 

3064 

2512 

1329 

8140 

45 

30 

8241 

5664 

6483 

1328 

4724 

6992 

2965 

2656 

1206 

8320 

30 

45 

8216 

5700 

6433 

1400 

4649 

7100 

2866 

2800 

1082 

8500 

15 

35  0 

8192 

5736 

6383 

1472 

4575 

7207 

2766 

2943 

0958 

8679 

55  0 

15 

0.8166 

0.5771 

1.6333 

1.1543 

2.4499 

.7314 

3.2666 

2.3086 

4.0832 

2.8857 

45 

30 

8141 

5807 

6282 

1614 

4423 

7421 

2565 

3228 

0706 

9035 

30 

45 

8116 

5842 

6231 

1685 

4347 

7527 

2463 

3370 

0579 

9212 

15 

36  0 

8090 

5878 

6180 

1756 

4271 

7634 

2361 

3511 

0451 

9389 

54  0 

15 

8064 

5913 

6129 

1826 

4193 

7739 

2258 

3652 

0322 

9565 

45 

30 

8039 

5948 

6077 

1896 

4116 

7845 

2154 

3793 

0193 

9741 

30 

45 

8013 

5983 

6025 

1966 

4038 

7950 

2050 

3933 

0063 

9916 

15 

37  0 

7986 

6018 

5973 

2036 

3959 

8054 

1945 

4073 

3.9932 

3.0091 

53  0 

15 

7960 

6053 

5920 

2106 

3880 

8159 

1840 

4212 

9800 

0265 

45 

30 

7934 

6088 

5867 

2175 

3801 

8263 

1734 

4350 

9668 

0438 

30 

45 

0.7907 

0.6122 

1.5814 

1.2244 

2.3721 

.8367 

3.1628 

2.4489 

3.9534 

3.0611 

15 

38  0 

7880 

6157 

5760 

2313 

3640 

8470 

1520 

4626 

9400 

0783 

52  0 

15 

7853 

6191 

5706 

2382 

3560 

8573 

1413 

4764 

9266 

0955 

45 

30 

7826 

6225 

5652 

2450 

3478 

8675 

1304 

4901 

9130 

1126 

30 

45 

7799 

6259 

5598 

2518 

3397 

8778 

1195 

5037 

8994 

1296 

15 

39  0 

7771 

6293 

5548 

2586 

3314 

8880 

1086 

5173 

8857 

1466 

51  0 

15 

7744 

6327 

5488 

2654 

3232 

8981 

0976 

5308 

8720 

1635 

45 

30 

7716 

6361 

5432 

2722 

3149 

9082 

0865 

5443 

8581 

1804 

30 

45 

7688 

6394 

5377 

2789 

3065 

9183 

0754 

5578 

8442 

1972 

15 

40  0 

7660 

6428 

5321 

2856 

2981 

9284 

0642 

5712 

8302 

2139 

50  0 

15 

0.7632 

0.6461 

1.5265 

1.2922 

2.2897 

.9384 

.0529 

2.5845 

3.8162 

3.2306 

45 

30 

7604 

6494 

5208 

2989 

2812 

9483 

0416 

5978 

8020 

2472 

30 

45 

7576 

6528 

5151 

3055 

2727 

9583 

0303 

6110 

7878 

2638 

15 

41  0 

7547 

6561 

5094 

3121 

2641 

9682 

0188 

6242 

7735 

2803 

49  0 

15 

7518 

6593 

6037 

3187 

2555 

9780 

0074 

6374 

7592 

2967 

45 

30 

7490 

6626 

4979 

3252 

2469 

9879 

.9958 

6505 

7448 

3131 

30 

45 

7461 

6659 

4921 

3318 

2382 

9976 

9842 

6635 

7303 

3294 

15 

42  0 

7431 

6691 

4863 

3383 

2294 

.0074 

9726 

6765 

7157 

3457 

48  0 

15 

7402 

6724 

4804 

3447 

2207 

0171 

9609 

6895 

7011 

3618 

45 

30 

7373 

6756 

4746 

3512 

2118 

0268 

9491 

7024 

6864 

3780 

30 

45 

0.7343 

0.6788 

1.4686 

.3576 

.2030 

.0364 

.9373 

2.7152 

.6716 

3.3940 

15 

43  0 

7314 

6820 

4627 

3640 

1941 

0460 

9254 

7280 

6568 

4100 

47  0 

15 

7284 

6852 

4567 

3704 

1851 

0555 

9135 

7407 

6419 

4259 

45 

30 

7254 

6884 

4507 

3767 

1761 

0651 

9015 

7534 

6269 

4418 

30 

45 

7224 

6915 

4447 

3830 

1671 

0745 

8895 

7661 

6118 

4576 

16 

44  0 

7193 

6947 

4387 

3893 

1580 

0840 

8774 

7786 

5967 

4733 

46  0 

15 

7163 

6978 

4326 

3956 

1489 

0934 

8652 

7912 

5815 

4890 

45 

30 

7133 

7009 

4265 

4018 

1398 

1027 

8530 

8036 

5663 

5045 

30 

45 

7102 

7040 

4204 

4080 

1306 

1120 

8407 

8161 

5509 

5201 

15 

45  0 

7071 

7071 

4142 

4142 

1213 

1213 

8284 

8284 

5355 

5355 

45  0 

e   < 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

•   * 

B'ng 

JMst.  1. 

IMs  t  .  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

B»n£ 

TABLE  V.   TRAVERSE  TABLE.          91 

,'B'ng 

Dist.  6. 

J»isi  7. 

Dist  .  8. 

Dist.  9.   Dist.  1O. 

B'ng 

.  . 

Lat 

Dep. 

Lat. 

Pep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat". 

Dep. 

.  . 

30  15 

5.1830 

3.0226 

6.0468 

3.5264 

6.9107 

4.0302 

7.7745 

4.5340 

8.6384 

5.0377 

5945 

30 

1698 

0452 

0314 

5528 

8930 

0603 

7547 

5678 

6163 

0754 

30 

45 

1564 

0678 

0158 

5791 

8753 

0903 

7347 

6016 

5941 

1129 

15 

31  0 

1430 

0902 

0002 

6053 

b573 

1203 

7145 

6353 

5717 

1504 

59  0 

15 

1295 

1126 

5.9844 

6314 

8393 

1502 

6942 

6690 

6491 

1877 

45 

30 

1158 

1350 

9685 

6575 

8211 

1800 

6738 

7025 

5264 

2250 

30 

45 

1021 

1573 

9525 

6835 

8028 

2097 

6532 

7359 

5035 

2621 

15 

32  0 

0883 

1795 

9363 

7094 

7844 

2394 

6324 

7693 

4805 

2992 

58  0 

15 

0744 

2017 

9201 

7353 

7658 

2689 

6116 

8025 

4573 

3361 

45 

30 

0603 

2238 

9037 

7611 

7471 

2984 

5905 

8357 

4339 

3730 

30 

45 

5.0462 

3.2458 

5.8873 

3.7868 

6.7283 

4.3278 

7.5694 

4.8688 

8.4104 

5.4097 

15 

33  0 

0320 

2678 

8707 

8125 

7094 

3571 

5480 

9018 

3867 

4464 

57  0 

15 

0177 

2898 

8540 

8381 

6903 

3863 

5266 

9346 

3629 

4829 

45 

30 

0033 

3116 

8372 

8636 

6711 

4155 

6050 

9674 

3389 

5194 

30 

45 

4.9888 

3334 

8203 

8890 

6518 

4446 

4832 

5.0001 

3147 

5557 

15 

34  0 

9742 

3552 

8033 

9144 

6323 

4735 

4613 

0327 

2904 

5919 

56  0 

15 

9595 

3768 

7861 

9396 

6127 

5024 

4393 

0652 

2659 

6280 

45 

30 

9448 

3984 

7689 

9648 

5930 

5312 

4171 

0977 

2413 

6641 

30 

45 

9299 

4200 

7515 

9900 

5732 

5600 

3948 

1300 

2165 

7000 

15 

35  0 

9149 

4415 

7341 

4.0150 

5532 

5886 

3724 

1622 

1915 

7358 

55  0 

15  4.8998 

3.4629 

5.7165 

4.0400 

6.5331 

4.6172 

7.3498 

5.1943 

8-1664 

5.7715 

45 

30 

8847 

4842 

6988 

0649 

5129 

6456 

3270 

2263 

1412 

8070 

30 

45 

8694 

5055 

6810 

0897 

4926 

6740 

3042 

2582 

1157 

8425 

15 

36  0 

8541 

5267 

6631 

1145 

4721 

7023 

2812 

2901 

0902 

8779 

54  0 

15 

8387 

5479 

6451 

1392 

4516 

7305 

2580 

3218 

0644 

9131 

45 

30 

8231 

5689 

6270 

1638 

4309 

7586 

2347 

3534 

0386 

9482 

30 

45 

8075 

5899 

6088 

1883 

4100 

7866 

2113 

3849 

0125 

9832 

15 

37  0 

7918 

6109 

5904 

2127 

3891 

8145 

1877 

4163 

7.9864 

6.0182 

53  0 

15 

7760 

6318 

5720 

2371 

3680 

8424 

1640 

4476 

9600 

0529 

45 

30 

7601 

6526 

5535 

2613 

3468 

8701 

1402 

4789 

9336 

0876 

30 

45 

4.7441 

3.6733 

5.5348 

4.2855 

6.3255 

4.8977 

7.1162 

5.5100 

7.9069 

6-1222 

15 

38  0 

7281 

6940 

5161 

3096 

3041 

9253 

0921 

6410 

8801 

1566 

52  0 

15 

7119 

7146 

4972 

3337 

2825 

9528 

0679 

5718 

8532 

1909 

45 

30 

6956 

7351 

4783 

3576 

2609 

9801 

0435 

6026 

8261 

2251 

30 

45 

6793 

7555 

4592 

3815 

2391 

5.0074 

0190 

6333 

7988 

2592 

15 

39  0 

6629 

7759 

4400 

4052 

2172 

0346 

6.9943 

6639 

7715 

2932 

51  0 

15 

6464 

7962 

4207 

4289 

1951 

0616 

9695 

6943 

7439 

3271 

45 

30 

6297 

8165 

4014 

4525 

1730 

0886 

9446 

7247 

7162 

3608 

30 

45 

6131 

8366 

3819 

4761 

1507 

1155 

9196 

7550 

6884 

3944 

15 

40  0 

5963 

8567 

3623 

4995 

1284 

1423 

8944 

7851 

6604 

4279 

50  0 

15 

4.5794 

3.8767 

5.3426 

4.5229 

6.1059 

5.1690 

6.8691 

5.8151 

7.6323 

6.4612 

45 

30 

5624 

8967 

3228 

5461 

0832 

1956 

8437 

8450 

6041 

4945 

30 

45 

5454 

9166 

3030 

5693 

0605 

2221 

8181 

8748 

5756 

5276 

15 

41  0 

5283 

9364 

2830 

5924 

0377 

2485 

7924 

9045 

5471 

5606 

49  0 

15 

5110 

9561 

2629 

6154 

0147 

2748 

7666 

9341 

5184 

6935 

46 

30 

4937 

9757 

2427 

6383 

5.9916 

3010 

7406 

9636 

4896 

6262 

30 

45 

4763 

9953 

2224 

6612 

9685 

3271 

7145 

9929 

4606 

6588 

15 

42  0 

4589 

4.0148 

2020 

6839 

9452 

3530 

6883 

6.0222 

4314 

6913 

48  0 

15 

4413 

0342 

1815 

7066 

9217 

3789 

6620 

0513 

4022 

7237 

46 

30 

4237 

0535 

1609 

7291 

8982 

4047 

6355 

0803 

3728 

7559 

30 

45 

4.4059 

4.0728 

5.1403 

4.7516 

5.8746 

5.4304 

6.6089 

6.1092 

7.3432 

6.7880 

15 

43  0 

3881 

0920 

1195 

7740 

8508 

4560 

5822 

1380 

3135 

8200 

47  0 

15 

3702 

1111 

0986 

7963 

8270 

4815 

5553 

1666 

2837 

8518 

45 

30 

3522 

1301 

0776 

8185 

8030 

5068 

5284 

1952 

2537 

8835 

30 

45 

3342 

1491 

0565 

8406 

7789 

5321 

5013 

22361  2236 

9151 

16 

44  0 

3160 

1680 

0354 

8626 

7547 

5573 

4741 

2519 

1934 

9466 

46  0 

15 

2978 

1867 

0141 

8845 

7304 

5823 

4467 

2801 

1630 

9779 

46 

30 

2795 

2055 

4.9928 

9064 

7060 

6073 

4193 

3082 

1325 

7.0091 

30 

45 

2611 

2241 

9713 

9281 

6815 

6321 

3917 

3361 

1019 

0401 

15 

45  0 

2426 

2426 

9497 

9497 

6569 

6569 

3640 

3640 

0711 

0711 

45  0 

•  ' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

•  ' 

iB'ng 

DIst.  6. 

DIst.  7. 

Dist.  8.  j  Dist.  9. 

Dist  10. 

B^S 

92  TABLE  VL      DEPABTUBE8, 

For  Correction  of  Courses  on  Random  LAnes. 


Minutes. 

10  Chains. 

20  Chains. 

40  Chains. 

80  Chains. 

Minutes. 

1 

.003 

.006 

.012 

.023 

l 

2 

006 

012 

023 

046 

2 

3 

009 

017 

035 

070 

3 

4 

012 

023 

046 

093 

4 

5 

014 

029 

058 

116 

5 

6 

017 

035 

070 

140 

6 

7 

020 

041 

081 

163 

8 

023 

046 

093 

186 

-  8 

9 

026 

052 

105 

209 

9 

10 

029 

058 

116 

233 

10 

11 

032 

064 

128 

256 

11 

12 

035 

070 

140 

279 

12 

13 

038 

076 

151 

302 

13 

14 

041 

081 

163 

326 

14 

15 

044 

087 

174 

349 

15 

16 

046 

093 

186 

372 

16 

17 

049 

099 

198 

396 

17 

18 

052 

105 

209 

419 

18 

19 

055 

110 

221 

442 

19 

20 

058 

116 

233 

466 

20 

21 

061 

122 

244 

488 

21 

22 

064 

128 

256 

512 

22 

23 

067 

134 

268 

535 

23 

24 

070 

140 

279 

558 

24 

25 

073 

145 

291 

581 

25 

26  - 

076 

151 

302 

605 

26 

27 

078 

157 

314 

628 

27 

28 

081 

163 

326 

651 

28 

29 

084 

169 

337 

674 

29 

30 

087 

174 

349 

698 

30 

31 

090 

180 

361 

722 

31 

32 

093 

186 

372 

744 

32 

33 

096 

192 

384 

767 

33 

34 

099 

198 

395 

790 

34 

35 

102 

204 

407 

814 

35 

36 

105 

209 

419 

837 

36 

37 

108 

215 

430 

860 

37 

38 

110 

221 

442 

883 

38 

39 

113 

227 

454 

906 

39 

40 

116 

233 

465 

929 

40 

41 

119 

238 

477 

953 

41 

42 

122 

244 

488 

976 

42 

43 

125 

250 

500 

999 

43 

44 

128 

256 

512 

1.022 

44 

45 

131 

262 

523 

.045 

45 

46 

134 

268 

535 

.068 

46 

47 

137 

273 

546 

.092 

47 

48 

140 

279 

558 

.115 

48 

49 

142 

285 

570 

.138 

49 

60 

145 

291 

581 

.161 

50 

51 

148 

297 

593 

1.184 

51 

52 

151 

302 

605 

1.207 

52 

53 

154 

308 

616 

1.230 

53 

54 

157 

314 

628 

1.253 

54 

55 

160 

320 

639 

1.276 

55 

56 

163 

326 

651 

1.299 

56 

57 

166 

331 

663 

1.323 

67 

58 

169. 

337 

674 

1.346 

58 

59 

172 

343 

686 

1.369 

69 

60 

174 

349 

698 

1.392 

60 

TABLE  VIL      NATURAL  SECANTS. 


1«  -  -  - 

11° 

!!•-- 

21C 

21*-- 

81- 

31'-- 

41- 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant 

Angle. 

Secant. 

1 

1.00015 

11 

1.01872 

21 

1.07115 

31 

1.16663 

10 

1.00021 

10 

1.01930 

10 

1.07235 

10 

1.16868 

20 

1.00027 

20 

1.01989 

20 

.07356 

20 

1.17075 

30 

1.00034 

30 

1.02049 

30 

.07479 

30 

1.17283 

40 

1.00042 

40 

1.02110 

40 

.07602 

40 

1.17493 

50 

1.00051 

50 

1.02171 

60 

.07727 

60 

1.17704 

2 

1.00061 

12 

1.02234 

22 

.07853 

32 

1.17918 

10 

1.00072 

10 

1.02298 

10 

.07981 

10 

1.18133 

20 

1.00083 

20 

1.02362 

20 

.08109 

20 

1.18350 

30 

1.00095 

30 

1.02428 

30 

.08239 

30 

1.18569 

40 

1.00108 

40 

1.02494 

40 

.08370 

40 

1.18790 

60 

1.00122 

60 

1.02562 

60 

.08503 

60 

1.19012 

3 

1.00137 

13 

1.02630 

23 

.08636 

33 

1.19236 

10 

1.00153 

10 

1.02700 

10 

.08771 

10 

1.19463 

20 

1.00169 

20 

1.02770 

20 

.08907 

20 

1.19691 

30 

1.00187 

30 

1.02842 

30 

.09044 

30 

1.19920 

40 

1.00205 

40 

1.02914 

40 

.09183 

40 

1.20152 

60 

1.00224 

60 

1.02987 

60 

.09323 

60 

1.20386 

4 

1.00244 

14 

1.03061 

24 

.09464 

34 

1.20622 

10 

1.00265 

10 

1.03137 

10 

.09606 

10 

1.20859 

20 

1.00287 

20 

1.03213 

20 

.09750 

20 

1.21099 

30 

1.00309 

30 

1.03290 

30 

.09895 

30 

1.21341 

40 

1.00333 

40 

1.03368 

40 

.10041 

40 

1.21584 

60 

1.00357 

60 

1.03447 

50 

.10189 

60 

1.21830 

5 

1.00382 

15 

1.03528 

25 

.10338 

36 

1.22070 

10 

1.00408 

10 

1.03609 

10 

.10488 

10 

1.22327 

20 

1.00435 

20 

1.03691 

20 

.10640 

20 

1.22579 

30 

1.00463 

30 

1.03774 

30 

.10793 

30 

1.22833 

40 

1.00491 

40 

1.03858 

40 

.10947 

40 

1.23089 

60 

1.00521 

60 

1.03944 

60 

.11103 

60 

1.23347 

6 

1.00551 

16 

1.04030 

26 

.11260 

36 

1.23607 

10 

1.00582 

10 

1.04117 

10 

.11419 

10 

1.23869 

20 

1.00614 

20 

1.04206 

20 

.11579 

20 

1.24134 

30 

1.00647 

30 

1.04295 

30 

.11740 

30 

1.24400 

40 

1.00681 

40 

1.04385 

40 

.11903 

40 

1.24669 

60 

1.00715 

60 

1.04477 

50 

.12067 

60 

1.24940 

7 

1.00751 

17 

1.04569 

27 

.12233 

37 

1.26214 

10 

1.00787 

10 

1.04663 

10 

.12400 

10 

1.25489 

20 

1.00825 

20 

1.04757 

20 

.12568 

20 

1.25767 

30 

1.00863 

30 

1.04853 

30 

.12738 

30 

1  26047 

40 

1.00902 

40 

1.04950 

40 

.12910 

40 

1  26330 

50 

1.00942 

60 

1.05047 

60 

.13083 

60 

1.26615 

8 

1.00983 

18 

1.05146 

28 

.13257 

38 

1.26902 

10 

1.01024 

10 

1.05246 

10 

.13433 

10 

1.27191 

20 

1.01067 

20 

1.05347 

20 

.13610 

20 

1.27483 

30 

1.01111 

30 

1.05449 

30 

.13789 

30 

1.27778 

40 

1.01155 

40 

1.05552 

40 

.13970 

40 

1.28075 

,      60 

1.01200 

60 

1.05657 

60 

.14152 

60 

1.28374 

9 

1.01247 

19 

1.05762 

99 

.14335 

39 

1.28676 

10 

1.01294 

10 

1.05869 

10 

.14521 

10 

1.28980 

20 

1.01342 

20 

1.05976 

20 

.14707 

20 

1.29287 

30 

1.01391 

30 

1.06085 

30 

.14896 

30 

1.29597 

40 

1.01440 

40 

1.06195 

40 

.15085 

40 

1.29909 

60 

1.01491 

50 

1.06306 

60 

.15277 

60 

1.30223 

10 

1.01543 

20 

1.06418 

30 

.15470 

40 

1.30541 

10 

1.01595 

10 

1.06531 

10 

.15665 

10 

1.30831 

20 

1.01649 

20 

1.06645 

20 

.15861 

20 

1.31183 

30 

1.01703 

30 

1.06761 

30 

.16059 

30 

1.31509 

40 

1.01768 

40 

1.06878 

40 

1.16259 

40 

1.31837 

60 

1.01815 

50 

1.06995 

60 

1.16460 

60 

1.32168 

94 


TABLE  VII.      NATUEAL  SECANTS. 


41° 

46» 

_jg» 

.  51° 

51° 

56° 

56° 

151  1 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

41 

1.32501 

46 

1.43956 

51 

1.58902 

56 

1.78829 

10 

1.32838 

10 

1.44391 

10 

1.59475 

10 

1.79604 

20 

1.33177 

20 

1.44831 

20 

1.60054 

20 

1.80388 

30 

1.33519 

30 

1.45274 

30 

1.60639 

30 

1.81180 

40 

1.33864 

40 

1.45721 

40 

.61229 

40 

1.81981 

50 

1.34212 

50 

1.46173 

50 

.61825 

50 

1.82790 

42 

1.34563 

47 

1.46628 

52 

.62427 

57 

1.83608 

10 

1.34917 

10 

1.47087 

10 

.63035 

10 

1.84435 

20 

1.35274 

20 

1.47551 

20 

.63648 

20 

1.85271 

30 

1.35634 

30 

1.48019 

30 

.64268 

30 

1.86116 

40 

1.35997 

40 

1.48491 

40 

.64894 

40 

1.86990 

50 

1.36363 

50 

1.48967 

50 

.65526 

50 

1.87834 

43 

1.36733 

48 

1.49448 

53 

.66164 

58 

1.88708 

10 

1.37105 

10 

1.49933 

10 

.66809 

10 

1.89591 

20 

.37481 

20 

1.50422, 

20 

.67460 

20 

1.90485 

30 

.37860 

30 

1.50916 

30 

.68117 

30 

1.91388 

40 

.38242 

40 

1.51415 

40 

.68782 

40 

1.92302 

50 

.38628 

50 

1.51918 

50 

.69452 

50 

1.93226 

44 

.39016 

49 

1.52425 

54 

.70130 

59 

.94160 

10 

.39409 

10 

1.52938 

10 

.70815 

10 

.95106 

20 

.39804 

20 

1.53455 

20 

.71506 

20 

.96062 

30 

.40203 

30 

1.53977 

30 

.72205 

30 

.97029 

40 

1.40606 

40 

1.54504 

40 

.72911 

40 

.98008 

50 

1.41012 

50 

1.55036 

50 

.73624 

50 

.98998 

45 

1.41421 

50 

1.55572 

55 

.74345 

60 

2.00000 

10 

1.41835 

10 

1.56114 

10 

.75073 

,10 

2.01014 

20 

1.42251 

20 

1.56661 

20 

.75808 

20 

2.02039 

30 

1.42670 

30 

1.57213 

30 

.76552 

30 

2.03077 

40 

1.43096 

40 

1.57771 

'  40 

.77303 

40 

2.04128 

50 

1.43524 

50 

1.58333 

50 

.78062 

50 

2.05191 

JAN.  1.  TABLE  VIII.  JAN.  1. 

AZIMUTHS  OP  POLARIS  AT  EXTREME  ELONGATIONS. 


3 

1906 

1907 

1908 

1909 

J9IO 

a 
J 

1906 

1907 

1908 

i  909 

I9IO 

25 

I  19  -I 

i  18. 

i  18.4 

i  18.1 

I  17-7 

50 

i  51-5 

I  5I.O 

I  5O.6 

i  50.1 

I  49-6 

26 

19.8 

19. 

19.1 

18.7 

18. 

51 

54-0 

53-5 

53-0 

52.5 

52.0 

27 

20.5 

20. 

19.8 

19.4 

19- 

52 

56.4 

55-9 

55-4 

54-9 

54.4 

28 

21.3 

2O. 

20.5 

20.1 

19- 

53 

59-1 

58.6 

58.1 

57  6 

57.1 

29 

22.1 

21. 

21.3 

20.g 

20. 

54 

2  O2.0 

2  OI.5 

2  OO.g 

2  OO.4 

59;9 

30 

22.8 

22. 

22.1 

21.7 

21. 

55 

O5.0 

04-4 

03-9 

03-4 

2  02.8 

31 

23.6 

23. 

22.9 

22.5 

22. 

56 

08.2 

07-7 

07-1 

06.6 

06.0 

32 

24-5 

24. 

23-8 

23-4 

23- 

57 

ii.  7 

II.  I 

10.5 

IO.O 

09.4 

33 

25-5 

25. 

24-7 

24-3 

24. 

58 

15-3 

14-7 

14.2 

13-6 

13.0 

34 

26.5 

26. 

25-7 

25-3 

25- 

59 

19.2 

18.6 

18.0 

17-4 

16.8 

35 

27-5 

27- 

26.8 

26.4 

26. 

60 

23-4 

22.8 

22.1 

21.5 

20.  Q 

36 

28.6 

28. 

27-9 

27-5 

27. 

61 

27-9 

27.1 

26.6 

25-9 

25.3 

37 

29.7 

29- 

29-0 

28.6 

28. 

62 

32-7 

32.1 

31-4 

30.8 

30.1 

38 

31-0 

30.6 

30.2 

29.8 

29- 

63 

38.0 

37-3 

36.6 

35-9 

35.2 

39 

32-3 

31.8 

31.4 

31.0 

3o. 

64 

43-6 

42.9 

42.2 

41-5 

40.8 

40 

33-6 

33.2 

32.8 

32.4 

32. 

65 

49-7 

49-0 

48.3 

47-5 

46.8 

41 

35-0 

34.6 

34-2 

33-8 

_  33- 

66 

56-3 

55-6 

54.8 

54-1 

53-3 

42 

36.5 

36.0 

35-6 

35-2 

34-8 

67 

3  03.6 

3  02.8 

3  02.0 

3  OI.2 

3  00.4 

43 

38.1 

37.6 

37-2 

36.8 

36.3 

68 

ii.  5 

10.7 

09.8 

09.0 

08.2 

44 

39-7 

39.2 

38.8 

38.4 

37-9 

69 

20.  i 

19-3 

18.4 

17.6 

16.7 

45 

41-4 

40.9 

40-5 

40.1 

39-6 

70 

-29-7 

28.8 

27-9 

27.0 

26.1 

46 

43-2 

42.7 

42.3 

41-9 

41.4 

7i 

40.3 

39-4 

38.4 

37-5 

36.5 

47 

45-1 

44.6 

44-2 

43-7 

43-3 

72 

52.1 

51-  1 

50.1 

49-1 

48.1 

48 

47-2 

46.7 

46.3 

45-8 

45-3 

49 

I  49-3 

i  48.8 

I  48.4 

I  47-9 

I  47-4 

TABLE  IX.      MULTIPLIERS  OF  R,  95 

For  one  revolutionof  Gradienter  Screw,  used  in  finding  d'  and  d.  Page  117. 


Elevation. 

Multipliers 
of  r. 

i 
Elevation. 

Multipliers 
of  r. 

Elevat'n. 

Multipliers 
of  r. 

Inc. 

Hor. 

' 

Inc. 

Hor. 

Inc. 

Hor. 

e. 

Dist. 

Dist. 

e. 

Dist. 

Dist. 

e. 

Dist. 

Dist. 

1    00 

99.97 

99.95 

o          / 

14 

96.79 

93.91 

•     t 
22    30 

92.01 

85.01 

2 

99.90 

99.84 

14    30 

96.56 

93.49 

23 

91.66 

84.37 

3 

99.81 

99.67 

15 

96.33 

93.05 

23    30 

91.31 

83.73 

4 

99.69 

99.44 

15    30 

96.09 

92.59 

24 

90.95 

83.08 

5 

99.53 

99.15 

16 

95.85 

92.13 

24    30 

90.58 

82.42 

6 

99.35 

98.80 

16    30 

95.60 

91.66 

25 

90.21 

81.75 

7 

99.13 

98.39 

17 

95  34 

91.17 

25    30 

89.83 

81.08 

8 

98.89 

97.92 

17    30 

95.07 

90.67 

26 

89.44 

80.39 

9 

98.61 

97.39 

18 

94.80 

90.15 

20    30 

89.05 

79.69 

10 

98.31 

96.81 

18    30 

94.52 

89.63 

27 

88.65 

78.99 

10    30 

98.14 

96.50 

19 

94.23 

89.09 

27    30 

88.24 

78.27 

11 

97.97 

96.17 

19    30 

93.93 

88.54 

28 

87.83 

77.55 

11    30 

97.79 

95.83 

20 

93.63 

87.97 

28    30 

87.40 

76.81 

12 

97.61 

95.47 

20    30 

93.32 

87.41 

29 

86.98 

76.07 

12    30 

97.41 

95.10 

21 

93.00 

86.82 

29    30 

86.54 

75.32 

13 

97.21 

94.72 

21    30 

92.68 

86.23 

30 

86.10 

74.67 

13    30 

97.00 

94.22 

22 

92.34 

85.61 

30    30 

85.66 

73.81 

TABLE  X.   ANGLES  OF  ELEVATION, 

Corresponding  to  numbers  of  Revolution  of  the  Gradienter  Screw, 


Screw. 

Angle. 

Screw. 

Angle. 

Screw. 

Angle. 

Rev.  Div. 

0           .             „ 

Rev.  Div. 

0               '           /' 

Rev.  Div. 

0 

0     1 

0  00    21 

0   10 

0      3    26 

1    00 

0    34    23 

2 

0    41 

20 

6    53 

2 

1    08    45 

3 

1    02 

30 

10    19 

3 

1    43    06 

1    23 

40 

13    45 

4 

2    17    26 

1    43 

50 

17    11 

5 

2    51    45 

2    04 

€0 

20    38 

6 

3    26    01 

2    24 

70 

24    04 

7 

4    00    15 

2    45 

80 

27    30 

8 

4    34    26 

9 

3    06 

90 

30    56 

9 

5    08    34 

0    10 

0     3    26 

1    00 

0    34    23 

10    00 

5    42    38 

TABLE   XI.      MEAN   REFRACTIONS, 

In  Declination,  for  use  with  Solar  Compass. 


i 

-< 
£ 

I 

Declinations. 

For  Latitude  30°. 

+2O° 

+  15° 

+  1O° 

+5° 

0° 

—5" 

—1O° 

—15° 

—  ao° 

Oh. 
2 
3 
4" 
5 

10" 
14 
20 
32 
I'OO 

15" 
19 
26 
39 
1  10 

21" 
25 
32 
46 
1'24 

27" 
31 
39 
52 
1'52 

33" 
38 
47 
1'06 
207 

40" 
46 
55 
110 
244 

48" 
54 
1'06 
1  35 
346 

57" 
1'05 
119 
1  57 
543 

ros" 

1  18 
136 
229 
1306 

For  Latitude  32°  30'. 

Oh. 
2 
3 

4 
5 

13" 
17 
23 
35 
1'03 

18" 
22 
29 
43 
1*15 

24" 
28 
35 
51 
1'31 

30" 
35 
43 
I'Ol 
1  53 

36" 
42 
51 
1*13 

220 

44" 
50 
I'Ol 
127 
305 

52" 
I'OO 
1  13 
1  46 

425 

1'02" 
1  11 

128 
213 
736 

114" 
1  26 
147 
254 

For  Latitude  35°. 

Oh. 
2 
3 
4 
5 

15" 
20 
26 
39 

1'07 

21" 
25 
33 

47 
1'20 

27" 
32 
39 
56 
1'38 

33" 
38 

47 
1'07 
200 

40" 
46 
56 
1'20 
234 

48" 
55 
1'07 
1  36 
329 

57" 
1'05 
121 
159 
514 

1'08" 
1  18 
138 
232 
1016 

1'21" 
I  35 
200 
325 

For  Latitude  37°  30'. 

Oh. 
2 
3 
4 

5 

18" 
22 
29 
43 
I'll 

24" 

28 
36 
51 

1'26 

30" 
35 
43 
I'Ol 
154 

36" 
42 
52 
1'13 
210 

44" 
50 
1'02 
127 
249 

52" 
I'OO 
114 
149 
355 

1'02" 
112 
129 
214 
615 

114" 
126 
149 
254 
1458 

1'29" 
145 
216 
405 

For  Latitude  40°. 

Oh. 
2 
3 

4 
5 

21" 
25 
33 

47 
115 

27" 
32 
40 
55 
1'31 

33" 
39 
48 
1'06 
151 

40" 
46 
57 
1'19 

220 

48" 
52 

ros 

136 
305 

57" 
106 
121 
158 
425 

1'08" 
119 
138 
230 
734 

1'21" 
1  35 
202 
321 

2518 

1'39" 
157 
236 
459 

For  Latitude  42°  30'. 

ill. 
2 
3 
4 
5 

24" 
28 
36 
50 
1'16 

30" 
35 
43 
I'OO 
136 

36" 
39 
52 
I'll 
1  58 

44" 
50 
1'02 
126 
230 

52" 
I'OO 
113 
144 
322 

1'02" 
1  12 
129 
210 
500 

114" 
126 
149 
249 
924 

1'29  ' 
145 
217 
355 

1'49" 
211 
259 
616 

For  Latitude  45°. 

Oh. 
2 
3 

4 
5 

27" 
32 
40 
54 
1'23 

33" 
39 

47 
104 
141 

40" 
46 
56 
1*16 
205 

48" 
52 
1'07 
133 
241 

57" 
106 
1  21 
154 
340 

1'08" 
1  19 
1  38 
224 
540 

1'21" 
135 
200 
311 
1202 

1'39" 
157 
234 
438 

2'02" 
229 
329 
81ft 

For  Latitude  47°  30" 

oh. 
2 
3 
4 
6 

30" 
35 
43 
56 
1'27 

36" 
42 
51 
1'09 
146 

44" 
50 

roi 

123 
212 

52" 
I'OO 
113 
140 
252 

1'02" 
1  12 
128 
205 
401 

114" 
126 
147 
240 
630 

1'29" 
145 
215 
339 
1619 

1'49" 
201 
256 
537 

218" 
251 
408 
1118 

TABLE  XH.      ACREAGE  OF  OPEN  DRAINS. 


97 


Showing  Number  of  Acres  served  try  drains  having  bottom  urfdth*  fron* 
1  ft  to  10  ft,  with  side  slopes  of  l  to  1,  on  the  supposition  of  1  Ineh  rain 
fatt  in  2t  hours,  one-half  of  which  reaches  the  drain. 
Computed  by  B.  F.  WELLES,  C.  K,  Marshall,  Mich. 


Fall  in  feet 
per 

Bottom  Widths. 

1ft. 

2ft. 

3ft. 

imi. 

100ft. 

3rd. 

2ft. 
deep. 

3ft 

deep. 

2ft. 
deep. 

3ft. 
deep. 

2ft, 
deep. 

3ft. 
deep. 

1.6 
2.0 
2.4 
2.8 
3.2 
36 
4.0 
4.8 
5.6 
6.4 
72 
8.0 

.030 
.038 
.045 
.053 
.060 
.070 
.076 
.091 
.110 
.120 
.136 
.150 

.04 
.05 
.06 
.07 
.08 
.09 
.10 
.12 
.14 
.16 
.18 
.20 

407 
462 
508 
553 
592 
631 
666 
733 
794 
•852 
905 
956 

981 
1105 
1218 
1319 
1416 
1505 
1590 
1748 
1895 
2030 
2154 
2273 

594 
665 
732 
797 
853 
939 
959 
1057 
1143 
1225 
1300 
1373 

1311 
1473 
1622 
1762 
1889 
2009 
2115 
2333 
2523 
2700 
2869 
3031 

780 
879 
968 
1053 
1128 
1198 
1264 
1391 
1499 
1612 
1715 
1809 

1649 
1861 
2047 
2217- 
2377 
2529 
2665 
2935 
3172 
.  3401 
3612 
3815 

Fall  in  Feet 
per 

Bottom  Widths. 

4ft. 

5ft. 

6ft. 

imi. 

100ft. 

8rd. 

2ft. 
deep. 

3ft. 

deep. 

2ft. 

deep. 

3ft. 

deep. 

2ft. 
deep. 

3ft. 
deep. 

1.6 
2.0 
2.4 
2.8 
3.2 
3.6 
4.0 
4.8 
5.6 
6.4 
7.2 
8.0 

.030 
.038 
.045 
.053 
.060 
.070 
.076 
.091 
.110 
.120 
.136 
.150 

.04 

.05 
.06 
.07 
.08 
.09 
.10 
.12 
.14 
.16 
.18 
.20 

976 
1094 
1206 
1308 
1404 
1494 
1579 
1731 
1878 
2013 
2137 
2256 

2003 
2249 
2477 
2684 
2872 
3049 
3227 
3553 
3849 
4115 
4372 
4609 

1171 
1316 
1448 
1572 
1684 
1790 
1894 
2089 
2257 
2415 
2566 
2705 

2357 
2650 
2910 
3158 
3384 
3598 
3800 
4173 
4512 
4833 
5141 
5412 

1368 
1541 
1699 
1835 
1970 
2097 
2211 
2436 
2632 
2820 
3001 
3165 

2716 
3046 
3362 
3642 
3908 
4150 
4322 
4810 
5203 
5571 
5927 
6257 

Fall  in  Feet 
per 

Bottom  Widths. 

j    7ft. 

8ft.   g 

10ft. 

imi. 

100ft. 

8rd. 

2ft. 
deep. 

3ft. 
deep. 

2ft. 

<teep. 

3ft. 

deep. 

2ft 

deep. 

3ft. 
deep. 

1.6 
2.0 
2.4 
2.8 
3.2 
3.6 
4.0 
4.8 
5.6 
6.4 
7.2 
8.0 

.030 
.038 
.045 
.053 
.060 
.070 
.076 
.091 
.110 
.120 
.136 
.150 

.04 
.05 
.06 
.07 
.08 
.09 
.10 
.12 
.14 
.16 
.18 
.20 

1574 
1768 
1946 
2115 
2259 
2403 
2538 
2792 
3029 
3240 
3443 
3629 

3074 
3469 
3807 
4131 
4427 
4695 
4963 
5443 
6894 
6317 
6715 
7078 

1767 
1983 
2181 
2369 
2538 
2697 
2848 
3130 
3393 
3628 
3854 
4070 

3458 
3877 
4265 
4622 
4948 
5258 
5552 
6094 
6591 
7067 
75CT 
7910 

2177 
2448 
2695 
2921 
3136 
3327 
3508 
3857 
4184 
4489 
4760 
5038 

4179 
4710 
5169 
5609 
6014 
6378 
6745 
7405 
8010 
8578 
9110 
9623 

FORMULAS:  v 


(  af  X  9000  1  ! 
1        P        \ 


-0.11 


A  =  Q  x  47.6  =  Acreage. 


98 


TABLE  Xin.      ACREAGE  OF  TILE  DRAINS. 


Showing  Number  of  Acres  drained  by  different  sizes  of  tile,  the  rainfall 
being  considered  as  equal  to  one-half  inch  in  depth  each  24  hours. 
Computed  by  R.  C.  CARPENTER,  Lansing,  Mich. 


Rate  of  Inclination. 

Acres  Drained. 

Feet  to  one  of  rise. 

2-in 
Tile. 

3-in. 
Tile. 

4-in. 
Tile. 

6-in. 
Tile. 

8-in. 
Tile. 

10-in. 
Tile. 

12-in. 
Tile. 

1  foot  in      10  feet 

6.6 

18.9 

1       "           20     " 

4.7 

13.0 

26  8 

* 

1       "           25     " 

4  2 

11  4 

24  0 

66  2 

"           30     »• 

3-9 

10.9 

21.9 

61.5 

126.4 

40      " 

3.4 

9.4 

19.0 

53.3 

109.6 

190.5 

, 

50      " 

3.0 

8.4 

17.0 

47.7 

98.0 

170.4 

269.0 

60      " 

2.7 

7.6 

15.6 

43.4 

90.0 

156.0 

246.0 

70      " 

2  5 

6.9 

14.5 

39.9 

83.0 

144.4 

228.1 

80      " 

2.3 

6.5 

13.4 

37.2 

77.0 

135.0 

213.0 

90      " 

2.2 

6.1 

12.6 

35.0 

72.5 

127.0 

200.5 

100      " 

2.0 

5.7 

11.9 

33.1 

69.2 

120.6 

190.5 

150      " 

1.6 

4.5 

9.5 

26.6 

56.0 

97.3 

154.4 

200      *' 

3.9 

8.2 

22.8 

48.0 

83.9 

132.5 

250      " 

3.5 

7.5 

20.4 

43.4 

74.4 

117.0 

300      »• 

6  9 

18  4 

38  2 

65  5 

107  0 

400      *' 

5.9 

16  5 

34  6 

60  3 

90  7 

500      " 

14.8 

30.1 

54.0 

81.6 

600      " 

13  3 

28  0 

48  6 

74.0 

800      " 

24  0 

41  9 

65  0 

1,000      " 

21.2 

37.2 

56.0 

1,500      " 



30.8 

47.0 

2,000 

40.8 

NOTE.—  Tile  should  not  be  laid  to  grades  where  numbers  are  re- 
placed by  dashes. 


TABLE  XIV.      CAPACITY  OF  TILE. 

Showing  carrying  capacity  of  different  sizes  of  tile,  in  gallons.    From, 
Catalogue  of  the  Bennett  Sewer  Pipe  Co.  ,  Jackson,  Mich. 


Carrying  Capacity— Gallons  per  Minute. 


4 

a- 


el 


& 

a* 

S3 


14 

21 

36 

54 

84 

144 

232 

267 

470 

830 

1300 

1760 

3000 


20 

30 

52 

78 

120 

208 

330 

378 

680 

1180 

1850 

2450 

4152 


28 
42 
76 
111 
169 
304 
470 


2630 
3450 


34 

52 

92 

134 

206 

368 

570 

655 

1160 

2040 

3200 

4180 

7202 


40 

60 

108 

159 

240 


1360 
2370 
3740 

4860 


74 

132 

192 

294 

528 

810 

926 

1670 

2920 

4600 


55 

85 

148 

219 


1340 
1920 
3340 
5270 
6850 
11743 


104 

184 

269 

414 

736 

1140 

1613 

2350 

4100 

6470 

8410 

14466 


TABLE  XV.      AZIMUTHS  OF  TANGENT. 


Lati- 
tude. 

imile. 

2  miles. 

Smiles. 

4  miles. 

30 

89  59  30 

c    >      •» 

89  58  59.9 

89  58  29.9 

89  67  69.9 

31 

59  28.8 

58  57.5 

58  26.3 

57  55.0 

32 

69  27.5 

58  55.0 

58  22.5 

67  60.0 

33 

59  26.2 

68  52.5 

58  18.7 

67  44.9 

34 

59  24.9 

68  49.9 

58  14.8 

67  39.7 

35 

69  23.6 

58  47.2 

58  10.8 

67  34.4 

36 

59  22.2 

68  44.4 

58  06.8 

67  28.9 

37 

59  20.8 

58  41.6 

58  02.5 

67  23.3 

38 

69  19.4 

58  38.8 

67  68.2 

67  17.5 

39 

69  17.9 

68  35.8 

67  63.7 

67  11.6 

40 

69  16.4 

58  32.8 

57  49.2 

67  05.5 

41 

69  14.8 

58  29.6 

57  44.4 

66  69.3 

42 

69  13.2 

58  26. 

67  39.6 

66  52.8 

43 

69  11.5 

58  23. 

57  34.6 

66  46.2 

44 

59  09.8 

58  19. 

67  29.5 

66  39.3 

45 

59  08.0 

58  16. 

57  24.1 

66  32.1 

46 

69  06.2 

58  12. 

57  18.6 

66  24.8 

47 

89  59  04.3 

89  58  08.6 

89  57  12.9 

89  56  17.1 

Lati- 
tude. 

5  miles. 

6  miles. 

7  miles. 

8  miles. 

30 

89  57  29.9 

89  56  59.8 

89  56  29.8 

89  55  69.8 

31 

57  23.8 

66  52.5 

66  21.3 

65  60.0 

32 

57  17.5 

56  45.0 

56  12.5 

65  40.0 

33 

57  11.2 

56  37.4 

66  03.6 

65  29.9 

34 

67  04.6 

66  29.6 

55  54.5 

65  19.4 

35 

66  68.0 

66  21.6 

55  45.2 

65  08.8 

36 

66  51.1 

66  13.4 

65  35.6 

64  57.8 

37 

66  44.1 

66  06.0 

65  25.8 

54  46.6 

38 

66  36.9 

65  56.3 

55  15.7 

64  36.1 

39 

66  29.6 

65  47.5 

55  05.4 

64  23.3 

40 

66  21.9 

55  38.3 

54  64.7 

64  11.1 

41 

66  14.1 

55  28.9 

64  43.7 

63  68.6 

42 

56  06.0 

55  19.2 

64  32.4 

63  46.6 

43 

65  57.7 

55  09.2 

64  20.8 

53  32.3 

44 

65  49.1 

64  68.9 

64  08.7 

63  18.5 

45 

65  40.2 

54  48.2 

53  56.3 

53  04.3 

46 

65  31.0 

54  37.2 

53  43.4 

52  49.5 

47 

89  55  21.4 

89  54  25.7 

89  53  30.0 

89  52  34.3 

lati- 
tude. 

Smiles. 

10  miles. 

11  miles. 

12  miles. 

o 

30 

89  65  29.8 

89  64  59.7 

89  54  29.7 

89  63  69.7 

31 

55  18.8 

64  47.6 

64  16.3 

63  46.1 

32 

65  07.6 

54  35.1 

54  02.6 

63  30.1 

33 

64  56.1 

64  22.3 

53  48.5 

63  14.8 

34 

54  44.4 

64  09.3 

53  34.2 

52  69.1 

35 

54  32.3 

63  55.9 

53  19.5 

62  43.1 

36 

64  20.0 

63  42.3 

53  04.5 

62  26.7 

37 

54  07.4 

63  28.2 

52  49.1 

52  09.9 

38 

63  64.5 

63  13.9 

62  33.2 

51  62.6 

39 

63  41.2  , 

52  59.1 

52  17.0 

51  34.9 

40 

63  27.5 

52  43.8 

52  00.2 

61  16.6 

41 

63  13.4 

62  28.2 

51  43.0 

60  57.8 

42 

52  58.8 

62  12.0 

51  25.2 

50  38.4 

43 

52  43.8 

51  55.4 

51  06.9 

50  18.5 

44 

52  28.4 

61  38.2 

50  48.0 

49  67.8 

45 

52  12.3 

51  20.4 

50  28.4 

49  36.4 

46 

61  55.7 

51  01.9 

50  08.1 

49  14.3 

47 

89  51  38.6 

89  50  42.9 

89  49  47.2 

89  48  51.4 

TABLE   XVI.      OFFSETS   FROM   TANGENT. 


Lati- 
tude. 

imile. 

2  miles. 

Smiles. 

4  miles. 

0 

Feet. 

Feet. 

Feet. 

Feet. 

30 

0.39 

.54 

3.47 

6.17 

31 

0.40 

.60 

3.61 

6.42 

32 

0.42 

.67 

3.76 

6.67 

33 

0.43 

.73 

3.90 

6.93 

34 

0.45 

.80 

4.05 

7.20 

36 

0.47 

.87 

4.20 

7.47 

36 

0.48 

.94 

4.36 

7.75 

37 

0.50 

2.01 

4.52 

8.04 

38 

0.52 

2.08 

4.69 

8.33 

89 

0.54 

2.16 

4.86 

8.63 

40 

0.56 

2.24 

5.03 

8.95 

41 

0.58 

2.32 

6.21 

9.27 

42 

0.60 

2.40 

6.40 

9.69 

43 

0.62 

2.48 

6.59 

9.93 

44 

0.64 

2.57 

6.79 

10.29 

45 

0.67 

2.66 

5.99 

10.65 

46 

0.69 

2.76 

6.20 

11.02 

47 

0.71 

2.85 

6.42 

11.41 

Lati- 
tude. 

5  miles. 

6  miles. 

7  miles. 

8  miles. 

0 

Feet. 

Feet. 

Feet. 

Feet. 

30 

9.64 

13.88 

18.89 

24.67 

31 

10.03 

14.44 

19.66 

25.68 

32 

10.42 

15.02 

20.44 

26.69 

33 

10.82 

15.60 

21.23 

27.74 

34 

11.25 

16.20 

22.06 

28.80 

35 

11.68 

16.81 

22.89 

29.89 

36 

12  11 

17.41 

23.74 

31.01 

37 

12.57 

18.09 

24.62 

32.16 

38 

13.02 

18.75 

25.52 

33.33 

39 

13.49 

19.43 

26.44 

34.54 

40 

13.98 

20.11 

27.40 

35.78 

41 

14.48 

20.85 

28.37 

37.06 

42 

14.99 

21.59 

29.38 

38.38 

43 

15.52 

22.35 

30.42 

39.74 

44 

16.07 

23.14 

31.50 

41.14 

45 

16.64 

23.96 

32.61 

42.59 

46 

17.21 

24.80 

33.  7S 

44.10 

47 

17.83 

25.68 

34.95 

45.65 

Lati- 
tude. 

9  miles. 

10  miles. 

11  miles. 

12  miles. 

. 

Feet 

Feet. 

Feet. 

Feet. 

30 

31.23 

38.55 

46.65 

55.52 

31 

32.49 

40.12 

48.54 

67.77 

32 

33.78 

41.71 

60.47 

60.06 

33 

35.10 

43.34 

62.44 

62.41 

34 

36.45 

45.00 

64.45 

64.80 

35 

37.83 

46.71 

66.62 

67.26 

36 

39.25 

48.45 

68.63 

69.77 

37 

40.70 

60.24 

60.79 

72.35 

38 

42.19 

62.08 

63.02 

75.00 

39 

43.71 

63.97 

65.30 

77.71 

40 

45.29 

65.91 

67.65 

80.51 

41 

46.90 

67.91 

70.07 

83.39 

42 

48.57 

69.97 

72.56 

86.35 

43 

60.29 

62.09 

75.13 

89.41 

44 

62.07 

64.28 

77.78 

92.57 

45 

63.91 

66.55 

80.53 

95.84 

48 

65.81 

68.90 

83.37 

99.22 

47 

57.78 

71.34 

86.32 

102.72 

TABLE  XVIL— Minutes  In  Decimals  jf  a  Be^rde.- 


1' 

.0167      11'    .1833      21' 

.3500 

31' 

5167 

41 

'    .6833 

61' 

.8500 

2 

.0333      12     .2000      22 

.3667 

32 

5333 

.7000 

62 

.8667 

3 

.0500      18     .2167      23 

.3833 

33 

5-500 

4J 

.7167 

63 

8833 

4 

.0667      14     .2333     24 

.4000 

34 

5667 

44 

.7333 

64 

.'9000 

6 

.0833      16     .2500      25 

.4167 

35 

5833 

45 

.7500 

66 

.9167 

ft 

.1000      16     .2667      26 

.4333 

36 

6000 

4fc 

.7667 

66 

.9333 

7 

.1167      17     .2833      27 

.4500 

37 

6167 

47 

.7833 

67 

.9500 

8 

.1333      18     .3000      28 

.4667 

38 

6333 

4S 

.8000 

68 

.9667 

9 

.1500      19     .3167      29 

.4833 

39 

.6500 

4< 

>     .8167 

69 

.9833 

10 

.1667      20     .3333      30 

.5<W» 

40 

6fifi7 

5( 

>     ,a333 

60 

1.0000 

TABLE  xvm.—  Inches  in  Decimals  of  a  Toot. 

1-16     3-32 

>£ 

3-16       M       5-16       % 

U 

% 

% 

% 

.0052    .0078 

.0104 

.0156    .0208    .0260    .0313    .0 

417 

.0521 

.0625 

.0729 

I           2 

3 

4567 

8 

9 

10 

11 

.0833    .1667 

.2500 

.3333    .4167    .5000    .5833    .6 

<J67 

.7500 

.8333 

.9167 

TABLE  xix.  —  Radii,  and  Deflections. 

Deg. 

Radius 

Tan. 
Def 

Chd. 
Def. 

Def. 
for 
I  Foot 

beg- 

Radios 

Tan. 
Def. 

Chd. 
Def. 

Def. 
for 
I  Pool 

o°  iy 

34377. 

.145 

.291 

0.05' 

7° 

819.0 

6.105 

12.21 

2.KX 

20 

17189. 

.291 

.582 

0.10 

20 

781 

4 

6.395 

12.79 

220 

30 

11459. 

.436 

.873 

0.15 

30 

764 

.5 

6.540 

13.08 

2.25 

40 

8594.4 

.582 

1.164 

0,20 

40 

747.9 

6.685 

13.37 

2.30 

50 

6875.5 

.727 

1.454 

0.25 

8 

716 

£ 

6.976 

13.95 

2.40 

1 

5729.6 

.873 

1745 

0.30 

20 

688 

.'2 

7.266 

14.53 

2.50 

10 

4911.2 

1.018 

2.036 

0.35 

30 

674 

.7 

7.411 

14.82 

2.55 

20 

4297.3 

1.164 

2.327 

0.40 

40 

661 

.7 

7.556 

15.11 

260 

30 

3819.8 

1.309 

2.618 

0.45 

9 

637.3 

7.846 

l-i69 

2.70 

40 

3437.9 

1.454 

2.909 

0.50 

20 

614 

.« 

8.136 

16.27 

2.80 

50 

3125.4 

1.600 

3.200 

0.55 

30 

6038 

8.281 

1656 

2.85 

2 

2864.9 

1.745 

8.490 

0.60 

40 

593 

.4 

8.426 

16.85 

2.90 

10 

2644.6 

1.891 

3.781 

0.65 

10 

573 

.7 

8.716 

17.43 

3.00 

20 

2455.7 

2.036 

4072 

0.70 

30 

546.4 

9.150 

18.30 

3.15 

30 

2292.0 

2.181 

4.363 

0.75 

11 

521 

7 

9.585 

19.16 

3.30 

40 

2148.8 

2.327 

4.6-54 

0.80 

30 

499 

.1 

10.02 

20.04 

3.45 

50 

2022.4 

2.472 

4945 

0.85 

12 

478 

3 

10.45 

20.91 

3.60 

8 

19101 

2.618 

5.235 

0.90 

30 

459 

.3 

10.89 

21.77 

3.75 

10 

1809.6 

2.763 

5.526 

0.95 

18 

441.7 

11.32 

22,64 

3.90 

20 

1719.1 

2.908 

5.817 

1.00 

30 

425 

4 

11.75 

23.51 

4.05 

30 

1637.3 

3.054 

6.108 

1.05 

14 

410.3 

12.18 

24.37 

4.20 

40 

1562.9 

3.199 

6.398 

1.10 

30 

396 

2 

12.62 

25.24 

4.35 

50 

1495.0 

3.345 

6.689 

1.15 

16 

383.1 

13.05 

26.11 

4.50 

4 

1432.7 

3490 

6.980 

1.20 

30 

370 

8 

13.49 

26.97 

4.65 

10 

1375.4 

3.635 

7.271 

1.25 

16 

359 

3 

13.92 

27.84 

4.80 

20 

1322.5 

3.781 

7.561 

1.30 

30 

348.5 

14.35 

28.70 

4.95 

30 

1273.6 

3926 

7.852 

1.35 

17 

338 

3 

14.78 

29.56 

5.10 

40 

1228.1 

4.071 

8.143 

1.40 

18 

319.6 

15.64 

31.29 

5.40 

50 

1185.8 

4.217 

8.433 

145 

19 

302 

9 

16.51 

33.01 

5.70 

6 

1146.3 

4.362 

8.724 

1.50 

20 

287 

9 

17.37 

34.73 

6.00 

10 

1109.3 

4.507 

9.014 

1.55 

21 

274.4 

18.22 

36.44 

6.30 

20 

1074.7 

4.653 

9.305 

1.60 

22 

262 

0 

19.08 

38.16 

6.60 

30 

1042.1 

4.798 

9.596 

1.65 

28 

250 

8 

1994 

39.87 

6.90 

40 

1011.5 

4.943 

9.886 

1.70 

24 

240.5 

20.79 

41.58 

7.20 

50 

982.6 

5.088 

10.18 

1.75 

26 

231 

0 

21.64 

43.28 

7.50 

6 

955.4 

5.234 

10.47 

1.80 

26 

222.3 

22.50 

44.99 

7.80 

10 

929.6 

5.379 

10.76 

1.85 

27 

214 

2 

23.35 

46.69 

8.10 

20 

905.1 

5.524 

11.05 

1.90 

28 

206 

7 

24.19 

48.38 

8.40 

30 

881.9 

5.669 

11.34 

1.95 

29 

199.7 

2504 

50.07 

8.70 

40 

859.9 

5.814 

11.63 

2.00 

80 

193.2 

25.88 

61.76 

9.00 

102  TABLE  xx— Tangents  and  Externals  to  a  1°  Curve. 


Angle. 

Tangent. 

Bxter'l. 

ingle. 

Tangent. 

External 

Angle. 

Tangent, 

External 

1° 

50.00 

.22 

11° 

551.70 

26.50 

21° 

1061.9 

97.57 

10' 

58.34 

.30 

10' 

560.11 

27.31 

W 

1070.6 

99.16 

20 

66.67 

.39 

20 

568.53 

28.14 

20 

10792 

100.75 

30 

7501 

.49 

30 

576.95 

2897 

30 

1087.8 

102.35 

40 

83.34 

.61 

40 

585.36 

29.82 

40 

1096.4 

10397 

50 

91.68 

.73 

50 

593.79 

30.68 

50 

1105.1 

105.60 

2 

100.01 

.87 

12 

602.21 

31.56 

22 

1113.7 

107.24 

10 

108.35 

1.02 

10 

610.64 

32.45 

10 

1122.4 

10890 

20 

116.68 

1.19 

20 

619.07 

33.35 

20 

1131.0 

110.57 

30 

125.02 

1.36 

30 

627.50 

34.26 

30 

1139.7 

11225 

40 

133.36 

1.55 

40 

635.93 

35.18 

40 

1148.4 

11395 

50 

141.70 

1.75 

50 

644.37 

36.12 

50 

1157.0 

115.66 

3 

150.04 

1.96 

13 

652.81 

3707 

23 

1165.7 

117.38 

10 

158.38 

2.19 

10 

661.25 

38.03 

10 

1174.4 

119.12 

20 

166.72 

2.43 

20 

669.70 

39.01 

20 

1183.1 

12087 

30 

175.06 

2.67 

30 

678.15 

39.99 

30 

1191.8 

122.63 

40 

183.40 

293 

40 

686.60 

40.99 

40 

1200.5 

124.41 

50 

191.74 

3.21 

50 

695.06 

42.00 

50 

1209.2 

126.20 

4 

200.08 

3.49 

14 

70351 

43.03 

24 

1217.9 

128.00 

10 

208.43 

3.79 

10 

711.97 

44.07 

10 

1226.6 

129.82 

20 

216.77 

4.10 

20 

720.44 

45.12 

20 

1235.3 

131.65 

30 

225.12 

4.42 

30 

728.90 

46.18 

30 

1244.0 

133.50 

40 

233.47 

4.76 

40 

737.37 

47.25 

40 

1252.8 

135.35 

50 

24181 

5.10 

50 

745.85 

48.34 

50 

1261.5 

137.23 

5 

250.16 

5.46 

15 

754.32 

49.44 

25 

1270.2 

13911 

10 

25851 

5.83 

10 

762.80 

50.55 

10 

1279.0 

141.01 

20 

266.86 

6.21 

20 

771.29 

51.68 

20 

12877 

142.93 

30 

275.21 

6.61 

30 

779.77 

52.89 

30 

1296.5 

144.85 

40 

283.57 

7.01 

40 

788.26 

53.97 

40 

1305.3 

146.79 

50 

291.92 

7.43 

50 

796.75 

55.13 

50 

1314,0 

148.75 

6 

300.28 

7,86 

16 

805.25 

5631 

26 

1322.8 

150  71 

10 

30864 

8.31 

10 

813.75 

57.50 

10 

1331.6 

152.69 

20 

316.99 

8.76 

20 

822.25 

58.70 

20 

1340.4 

154.69 

30 

325.35 

9.23 

30 

830.76 

59.91 

30 

1349.2 

156.70 

40 

333.71 

971 

40 

839.27 

61.14 

40 

1358.0 

158.72 

50 

342.08 

10.20 

50 

847.78 

62.38 

50 

1366.8 

160.76 

7 

350.44 

10.71 

17 

856.30 

63.63 

27 

1375.6 

162.81 

10 

358.81 

11.22 

10 

864.82 

64.90 

10 

1384.4 

164.86 

20 

367.17 

11.75 

20 

873.35 

66.18 

20 

1393.2 

166.95 

30 

375.54 

12.29 

30 

881.88 

67.47 

30 

1402.0 

169.04 

40 

383.91 

12.85 

40 

890.41 

68.77 

40 

1410.9 

171.15 

50 

392.28 

13.41 

50 

898.95 

70.09 

50 

1419.7 

173.27 

8 

400.66 

13.99 

18 

907.49 

71.42 

28 

1428.6 

175.41 

10 

409.03 

14.58 

10 

916,03 

72.76 

10 

1437.4 

177.55 

20 

417.41 

15.18 

20 

924.58 

74.12 

20 

1446.3 

179.72 

30 

425.79 

15.80 

30 

933.13 

75,49 

30 

1455.1 

181.89 

40 

434.17 

L6.43 

40 

941.69 

76.86 

40 

14640 

184.08 

50 

442.55 

17.07 

50 

950.25 

78.26 

50 

1472.9 

18629 

9 

450.93 

17.72 

19 

958.81 

79.67 

29 

1481.8 

188.51 

10 

45932 

18.38 

10 

967.38 

81.09 

10 

1490.7 

190.74 

20 

467.71 

19.06 

20 

975.96 

82.53 

20 

1499.6 

192.99 

30 

476.10 

19.75 

30 

984.53 

83.97 

30 

1508.5 

195,25 

40 

484.49 

20.45 

40 

993.12 

85.43 

40 

1517.4 

197.53 

50 

492.88 

21.16 

50 

1001.7 

86.90 

50 

1526.3 

199.82 

10 

501.28 

21.89 

20 

10103 

88.39 

80 

1535.3 

20212 

10 

509.68 

22.62 

10 

1018.9 

89.89 

10 

1544.2 

204.44 

20 

518.08 

23.38 

20 

1027.5 

91.40 

20 

1553.1 

206.77 

30 

526.48 

24.14 

30 

1036.1 

92.92 

30 

1562.1 

209.12 

40 

53489 

24.91 

40 

1044.7 

94.46 

40 

1571.0 

211.48 

60 

513.29 

25.70 

50 

1053.3 

96.01 

50 

1580,0 

213,86 

TABLE  xx— Tangents  and  Externals  to  a  1°  Curve.  103 


ingle. 

Tangent 

Exter'l. 

ingle. 

Tangent. 

External 

ingle. 

Tangent. 

Extarnal 

81° 

1589.0 

216.3 

41° 

2142.2 

387.4 

61° 

2732.9 

618.4 

i<y 

1598.0 

218.7 

10* 

2151.7 

390.7 

W 

2743.1 

622.8 

20 

1606.9 

221.1 

20 

2161.2 

394.1 

20 

2753.4 

627.2 

30 

1615.9 

223.5 

30 

2170.8 

397.4 

30 

2763.7 

631.7 

40 

16249 

226.0 

40 

2180.3 

400.8 

40 

2773.9 

636.2 

50 

1633.9 

223.4 

50 

2189.9 

404.2 

60 

2784,2 

640.7 

82 

1643.0 

230.9 

42 

2199.4 

407.6 

62 

2794.5 

646.2 

10 

1652.0 

233.4 

10 

2209.0 

411.1 

10 

2804.9 

649.7 

20 

1661.0 

235.9 

20 

2218.6 

414.5 

20 

2815.2 

6M.3 

30 

1670.0 

238.4 

30 

2228.1 

418.0 

30 

2825.6 

658.8 

40 

1679.1 

241.0 

40 

2237.7 

421.4 

40 

2835.9 

663.4 

50 

1688.1 

243.5 

50 

2247.3 

425.0 

50 

2846.3 

668.0 

33 

1697.2 

2461 

43 

2257.0 

428.5 

53 

2856.7 

6727 

10 

1706.3 

248.7 

10 

2266.6 

432.0 

10 

2867.1 

677.3 

20 

1715.3 

251.3 

20 

2276.2 

435.6 

20 

2877.5 

682.0 

30 

1724.4 

253.9 

30 

2235.9 

439.2 

30 

2888.0 

686-7 

40 

1733.5 

256.5 

40 

2295.6 

442.8 

40 

2>98.4 

691.4 

50 

1742.6 

259.1 

50 

2305.2 

446.4 

50 

2908.9 

696.1 

34 

1751.7 

261.8 

44 

2314.9 

450.0 

54 

2919.4 

700.9 

10 

1760.8 

264.5 

10 

2324.6 

453.6 

10 

2929.9 

705.7 

20 

1770.0 

267.2 

20 

2334.3 

457.3 

20 

2940.4 

710.5 

30 

1779  1 

2699 

30 

2344.1 

461.0 

30 

2951.0 

715.3 

40 

1788.2 

272,6 

40 

2353.8 

464.6 

40 

2961.5 

720.1 

50 

1797.4 

275.3 

50 

2363.5 

468.4 

50 

2972.1 

725.0 

35 

1806.6 

278.1 

45 

2373.3 

472.1 

55 

2982.7 

729.9 

10 

1815.7 

280.8 

10 

2383,1 

475.8 

10 

2993.3 

734.8 

20 

1824.9 

283.6 

20 

2392.8 

479.6 

20 

3003.9 

739.7 

30 

1834.1 

286.4 

30 

2402.6 

483.8 

30 

3014.5 

744.6 

40 

1843.3 

289.2 

40 

2412.4 

487.2 

40 

30i5.2 

749.6 

50 

1852.5 

292.0 

50 

2422.3 

491.0 

50 

3035.8 

754.6 

86 

1861.7 

2949 

46 

2432.1 

494.8 

56 

3046.5 

7596 

10 

1870.9 

297.7 

10 

2441.9 

498.7 

10 

3057.2 

7646 

20 

1880.1 

300.6 

20 

2451.8 

502.5 

20 

3067.9 

769.7 

30 

1889.4 

303.5 

30 

2461.7 

506.4 

30 

3078.7 

774.7 

40 

1898.6 

306.4 

40 

2471.5 

510.3 

40 

3089.4 

779.8 

50 

1907.9 

309.3 

50 

2481.4 

514.3 

60 

3100.2 

784.9 

37 

1917.1 

312.2 

47 

2491.3 

518.2 

57 

3110.9 

790.1 

10 

1926.4 

315.2 

10 

2501.2 

622.2 

10 

3121.7 

795.2 

20 

19&5.7 

318.1 

20 

2511.2 

526.1 

20 

3132.6 

800.4 

30 

1945.0 

321.1 

30 

2521.1 

530.1 

30 

3143.4 

805.6 

40 

1954.3 

324.1 

40 

2531.1 

534.2 

40 

3154.2 

810.9 

50 

1963.6 

327.1 

50 

2541.0 

538.2 

50 

3165.1 

816.1 

88 

1972.9 

330.2 

48 

2651.0 

542.2 

58 

3176.0 

821.4 

10 

1982.2 

333.2 

10 

25610 

546.3 

10 

3186.9 

826.7 

20 

1991.5 

336.3 

20 

2571.0 

550.4 

20 

3197.8 

832.0 

30 

2000.9 

339.3 

30 

2581.0 

554.5 

30 

3208.8 

837.3 

40 

2010.2 

342.4 

40 

2591.0 

558.6 

40 

3219.7 

842.7 

50 

2019.6 

345.5 

50 

2601.1 

-662.8 

50 

3230.7 

848J 

89 

2029.0 

348.6 

49 

2611.2 

566.9 

69 

3241.7 

853.5 

10 

2038.4 

351.8 

10 

2621.2 

571.1 

10 

3252.7 

858.9 

20 

2047.8 

3549 

20 

2631.3 

575.3 

20 

3263.7 

864.3 

30 

2057.2 

358.1 

30 

2641.4 

579.5 

30 

3274.8 

869.8 

40 

2066.6 

361.3 

40 

2651.5 

583.8 

40 

3285.8 

875.3 

60 

2076.0 

364.5 

50 

2661.6 

688.0 

60 

3296.9 

880.8 

40 

2085.4 

367.7 

50 

2671.8 

592.3 

60 

3308.0 

886.4 

10 

2094.9 

371.0 

10 

2681.9 

596.6 

10 

3319.1 

892.0 

20 

2104.3 

3742 

20 

2692.1 

600.9 

20 

3330.3 

897.5 

30 

2113.8 

377.5 

30 

2702.3 

605.3 

30 

3341.4 

903.a 

40 

2123.3 

380.8 

40 

2712.5 

609.6 

40 

3352.6 

908.8 

50 

2132.7 

384.1 

50 

2722.7 

614.0 

50 

3363.8 

914.5 

104  TABLE  xx— Tangents  and  Externals  to  a  1°  Curve. 


Ingle 

Tangent 

Externa 

Angle 

Tangent 

Externa 

Angle. 

Tangent 

External 

61° 

10 
20 
30 
40 
50 

3375.0 
3386.3 
3397.5 
3408.8 
3420.1 
3431.4 

920.2 
925.9 
931.6 
937.3 
943.1 
948.9 

71° 

10 
20 
30 
40 
50 

4086.9 
4099.5 
4112.1 
4124.8 
4137.4 
4150.1 

1308.2 
1315.6 
1322.9 
1330.3 
1337.7 
1345.1 

81° 
10' 
20 
30 
40 
50 

4893.6 
4908.0 
4922.5 
4937.0 
4951.5 
4966.1 

1805.3 
1814.7 
1824.1 
1833.6 
1843.1 
1852.6 

62 

10 

20 
30 
40 
50 

3442.7 
3454.1 
3465.4 
3476.8 
34883 
3499.7 

954.8 

960.6 
966.5 
972.4 
978.3 
984.3 

72 

10 
20 
30 
40 
50 

-4162.8 
4175.6 
4188.5 
4201.2 
4214.0 
4226.8 

1352.6 
1360.1 
1367.6 
1375.2 

1382.8 
1390.4 

82 
10 
20 
30 

40 
50 

4980.7 
4995.4 
5010.0 
5024.8 
5039.5 
5054.3 

1862.2 
1871.8 
1881.5 
1891.2 
190C9 
1910,7 

63 

10 

3511.1 
3522.6 

990.2 
996.2 

78 

10 

4239.7 
4252.6 

1398.0 
1405.7 

83 

10 

5069.2 
50840 

1920.5 
1930.4 

20 
30 

3534.1 
3545.6 

10023 
100S.3 

20 
30 

4265.6 

4278.5 

1413.5 
1421.2 

20 
30 

5099.0 
5113.9 

1940!3 
1950  3 

40 
50 

3557.2 

3568.7 

1014.4 
1020.5 

40 
50 

4291.5 
4304.6 

1429.0 
1436.8 

40 

50 

5128.9 
5143.9 

1980.2 

1970.3 

64 

3580.3 

1026.6 

74 

4317.6 

1444  6 

84 

5159  0 

1980.4 

10 

3591.9 

1032.8 

10 

4330.7 

1452.5 

10 

5174  1 

20 

3603.5 

1039  0 

20 

4343.8 

1460.4 

20 

61893 

2000.'6 

3J 

3615.1 

1045.2 

30 

4356.9 

1468.4 

30 

5204.4 

20108 

40 

50 

3626.8 
3638.5 

1051.4 
1057.7 

40 
50 

4370.1 
4383.3 

1476.4 
1484.4 

40 

50 

5219.7 
5234.9 

202U 
2031.4 

65 

10 
20 
30 
40 

50 

3650.2 
3661.9 
3673,7 
3685.4 
3697.2 
3709.0 

1063,9 
1070.2 
1076.6 
1082.9 
1089.3 
1095.7 

75 

10 
20 
30 
40 
50 

4396.5 
4409.8 
4423.1 
4436.4 
4449.7 
4463.1 

1492.4 
1500.6 
1508.6 
1516.7 
1524.9 
1533.1 

85 
10 
20 
30 
40 
50 

5250.3 
5265.6 
5281.0 
5296.4 
5311.9 
6327.4 

2041.7 
2052.1 
2062.5 
2073.0 
2083.5 
2094.1 

66 

3720.9 

1102.2 

76 

4476.5 

1541.4 

86 

5343.0 

2104.7 

10 

20 
30 
40 
50 

3732.7 
3744.6 
3756.5 

3768.5 
3780.4 

1108.6 
1115.1 
1121.7 
1128.2 
1134.8 

10 
20 
30 
40 
50 

4489.9 
4503.4 
4516.9 
4530.4 
4544.0 

1549.7 
1558.0 
1566.3 
1574.7 
1583.1 

10 
20 
30 
40 
50 

5358.6 
5374.2 
5389.9 
5405.6 
5121.4 

2115.3 
21260 
2136.7 
2147.5 

2158.4 

67 

10 
20 

3792.4 
3804.4 
38164 

1141.4 
1148.0 
1154.7 

77 
10 
20 

4557.6 
4571.2 
4584.3 

1591.6 
1600.1 
16086 

87 
10 
20 

5437.2 
5453.1 
54690 

2169.2 
2180.2 
2191  1 

30 
40 
60 

3828.4 
3840.5 
3852.6 

1161.3 
1168.1 
1174.8 

30 
40 
60 

4598.5 
4612.2 
4626.0 

1617.1 
1625.7 
1634.4 

30 
40 
50 

5484.9 
5500.9 
5517.0 

2202,2 
2213.2 
2224.3 

68 
10 

3864.7 
3876,8 

4181.6 
1188.4 

78 
10 

4639.8 
4653.6 

1643.0 
1651.7 

88 
10 

5533.1 

5549.2 

2235.5 
2246.7 

20 

3889.0 

1195.2 

20 

4667.4 

16605 

20 

5565.4 

2258  0 

30 

3901.2 

1202.0 

30 

4681.3 

1669.2 

30 

5581  6 

22693 

40 

50 

3913.4 
3925.6 

1208.9 
1215.8 

40 
50 

4695.2 
4709.2 

1678.1 
1686.9 

40 

50 

5597^8 
5614.2 

2280^6 
2292.0 

69 

3937.9 

1222.7 

79 

4723.2 

1695.8 

89 

56305 

23035 

10 

20 

3950.2 
3962.5 

1229.7 
1236.7 

10 

20 

4737.2 
4751.2 

1704.7 
17137 

10 
20 

5646.9 
.'j663.4 

2315.0 
23266 

30 
40 

3974.8 
3987.2 

1243.7  , 

1250.8 

30 
40 

4765.3 
4779.4 

1722.7 
1731.7 

30 
40 

5679.9 
5696.4 

2388.2 

2349  8 

50 

3999.5 

1257.9 

50 

4793.6 

1740.8 

60 

5713.C 

236L5 

70 

4011.9 

1265.0 

80 

4807.7 

1749.9 

90 

5729.7 

23733 

10 
20 

4024.4 
4036.8 

1272.1 
1279.3 

10 
20 

4822.0 
4836.2 

1759.0 
1768.2 

10 
20 

5746.3 
5763.1 

2385.1 
2397  0 

130 

4049.3 

1286.5 

30 

4850.5 

1777.4 

30 

57799 

2408.9 

40 

4061.8 

1293.6 

40 

4864.8 

17*6.7 

40 

6796.7 

24209 

60 

4074.4 

1300.9 

50 

4879.2 

1796.0 

50 

5813.6 

2432.9 

TABLE  xx— Tangents  and  Externals  to  a  1°  Curve.  105 


Ingle. 

Tangent    External 

Angle. 

Tangent 

External    '   Angle. 

Tangent 

External 

91° 

583U.5 

2444.9 

101° 

6950.6 

3278.1 

111° 

8336.7 

4386.1 

Iff 

5847.5 

2457.1 

10' 

6971.3 

3294.1 

HX 

8362.7 

4407.6 

20 

5864.6 

2469.3 

20 

6992.0 

3310.1 

20 

83SS.9 

4429.2 

30 

5881.7 

2481.5 

30 

7012.7 

£326.  1 

30 

8415.1 

44509 

40 

5898.8 

2493.8 

40 

7033.6 

3342.3 

40 

8441.5 

4472.7 

60 

5916.0 

2506.1 

50 

7054.5 

3358.5 

60 

846j.O 

4494.6 

92 

5933.2 

2518.5 

102 

7075.5 

3374.9 

112 

8494.6 

4516.6 

10 

59-50.5 

2531.0 

10 

7096.6 

3391.2 

10 

6521.3 

4538.8 

20 

5967.9 

2543.5 

20 

7117.8 

3407.7 

20 

8548.1 

4561.1 

30 

5985.3 

2556.0 

30 

7139.0 

3124.3 

30 

8575.0 

45^3.4 

40 

6'  ii)2.7 

2-568.6 

40 

7160.3 

3440.9 

40 

8602.1 

4606.0 

50 

6020.2 

2581.3 

5'J 

7181.7 

3157.6 

5J 

8629.3 

4628.6 

93 

6037.8 

2594.0 

103 

7203.2 

3474.4 

113 

8656.6 

4651.3 

10 

6055.4 

26)68 

10 

7224.7 

3491.3 

10 

8684.0 

4674.2 

20 

6073.1 

2619.7 

20 

7246.3 

3508.2 

20 

8711.5 

4697.2 

30 

6090.8 

2632.6 

30 

7268.0 

3525.2 

30 

8739.2 

4720.3 

40 

6108.6 

2645.5 

40 

72898 

3542.4 

40 

8767.0 

4743.6 

50 

6126.3 

2658.5 

50 

7311.7 

3559.6 

60 

8794.9 

4766-9 

94 

6144.3 

2671.6 

104 

73336 

3576.8 

114 

8822.9 

4790.4 

10 

6162.6 

26S4.7 

10 

7355.6 

£594.2 

10 

SS51.0 

4814.1 

20 

6180.2 

2697.9 

20 

7377.8 

3611.7 

20 

8879.3 

4887.8 

30 

6198.3 

2711.2 

30 

7399.9 

3629.2 

30 

8907.7 

4861.7 

40 

6216.4 

2724.5 

40 

7422.2 

3646.8 

40 

8936.3 

4885.7 

50 

6234.6 

2737.9 

50 

7444.6 

3664.5 

60 

8965.0 

4909.9 

95 

6252.8 

2751.3 

105 

7467.0 

3682.3 

115 

8993.8 

4934.1 

10 

6271.1 

2764,8 

10 

74896 

3700.2 

10 

9022.7 

4958.6 

20 

6289.4 

2778.3 

20 

7.512.2 

3718.2 

20 

9051.7 

4983.1 

30 

6307.9 

2792.0 

30 

7534  9 

3736.2 

30 

9080.9 

5007.8 

40 

6326.3 

2805.6 

40 

75-57.7 

3754.4 

40 

9110.3 

.5032.6 

50 

6344.8 

2819.4 

50 

75o0.5 

3772.6 

50 

9139.8 

6057.6 

96 

6363.4 

2833.2 

106 

7603.5 

3791.0 

116 

9169.4 

5082.7 

10 

6382.  1 

2847.0 

10 

7626.6 

3809.4 

10 

9199.1 

5107.9 

20 

6400.8 

2861.0 

20 

7649.7 

3827.9 

20 

9229.0 

5133.3 

30 

6419.5 

2875.0 

30 

7672,9 

3846.5 

30 

9259.0 

51-58.8 

40 

6438.4 

2889!o 

40 

7696.3 

3*65.2 

40 

9289.2 

5184.5 

50 

6457.3 

2903.1 

50 

7719.7 

38*4.0 

50 

9319.5 

6210.3 

97 

6476.2 

2917.3 

107 

7743,2 

3902.9 

117 

9349.9 

5236.2 

10 

6495.2 

2931.6 

10 

77668 

8921.9 

10 

9380.5 

5262.3 

20 

6514.3 

2945.9 

20 

7790.5 

3940.9 

20 

9411.3 

5288.6 

30 

6.533.4 

2960.3 

30 

7814.3 

3960.1 

30 

9442.2 

5315.0 

40 

6-552.6 

2974.7 

40 

7838-1 

3979.4 

40 

9473.2 

5341.5 

50 

6571.9 

2989.2 

60 

7862.1 

3998.7 

50 

9504.4 

5368.2 

98 

6591.2 

3003.8 

108 

7886.2 

4018.2 

118 

9535.7 

6395.1 

10 

6610.6 

3018.4 

10 

791U.4 

4037.8 

10 

9567.2 

5422.1 

20 

6630.1 

3033.1 

20 

7934.6 

4057.4 

20 

9598.9 

5449.2 

30 

6649.6 

3047.9 

30 

79.50.0 

4077.  '_' 

30 

9630.7 

5476.5 

40 

6669.2 

3062.8 

40 

7983.5 

4097.1 

40 

9662.6 

5504.0 

60 

6688.8 

3077.7 

50 

8008.0 

4117.0 

50 

9694.7 

5531.7 

99 

6708.6 

3092.7 

109 

8032.7 

4137.1 

119 

9727.0 

5559.4 

10 

6728.4 

3107.7 

10 

8057.4 

4157.3 

10 

9759.4 

5587.4 

20 

6748.2 

3122.9 

20 

&H2.3 

4177.5 

20 

9792.0 

5615.5 

30 

6768.1 

3138.1 

30 

8107.3 

4197.9 

30 

9824.8 

5613.8 

40 

67HS.1 

3153.3 

40 

81323 

4218.4 

40 

9857.7 

5672.3 

50 

6808.2 

3168.7 

60 

8157.5 

4239.0 

60 

9890.8 

5700.9 

100 

6828.3 

3184.1 

110 

8182.8 

4259.7 

120 

9924.0 

5729.7 

10 

6848.5 

3199.6 

10 

8208.2 

4280.5 

10 

9957.5 

5758.6 

20 

8B8&8 

3215.1 

20 

8233.7 

4301.4 

20 

9991.0 

5787.7 

30 

6889.2 

3230.8 

30 

8259.3 

4322.4 

30 

10025.0 

5817.0 

40 

6909.S 

3246.5 

40 

8285.0 

4343.6 

40 

lix  1.59.0 

5846.5 

50 

6930.1 

3262,3 

60 

8310.8 

4364.8 

60 

10093.0 

6876.1 

106 


TABLE    XXI.        STADIA    REDUCTIONS 


BY  ARTHUR    WINSLOW 
STADIA   REDUCTIONS   FOR   READING    100 


0° 

1° 

2° 

3° 

Minutes. 

Hor.      Diff. 

Hor.      Diff. 

Hor.      Diff. 

Hor.      Diff. 

Dist.     Elev. 

Dist.     Elev. 

Dist.     Elev. 

Dist.     Elev. 

0' 

100.00      .00 

99.97      1.74 

99.88      3.49 

99.73      5.23 

2 

.06 

.80 

99.87      3.55 

99.  73      5.28 

4 

.12 

"           .86 

3.60 

99.71      5.34 

6 

.17 

99.96        .92 

3.66 

44         5.40 

8 

.23 

.98 

99.86      3.72 

99.70      5.46 

10 

.29 

"         2.04 

3.78 

99.69      5,52 

12 

.35 

2.09 

99".  85      3.84 

"         5.57 

14 

.41 

99.95      2.15 

"         3.90 

99.68      5.63 

16 

«'         .47 

"         2.21 

99.84      3.95 

5.69 

18 

"         .52 

"         2.27 

.01 

99.67      5.76 

20 

.58 

'*         2.33 

99.83        .07 

99.66     5.80 

22 

"         .64 

99.94     2.38 

.13 

14         5.«6 

24 

"         .70 

44         2.44 

99.82        .18 

99.65      5.92 

26 

99.99      .76 

"         2.50 

"            .24 

99.64      5.98 

28 

.81 

99.93      2,56 

99.81        .30 

99.63      6.04 

30 

.87 

44      '  2.62 

.36 

44         6.09 

32 

.93 

"         2.67- 

99.80      4.42 

99^62      6.15 

34 

.99 

2.73 

44         4.48 

6.21 

36 

"       1.05 

99.92      2.79 

99.79      4.53 

99.61      6.27 

38 

li       1.11 

"         2.85 

"-"      4.59 

99.60      6.38 

40 

"       1.16 

2.91 

99.78      4.65 

99.59      6.38 

42 

"          .22 

99.91      2.97 

4.71 

a  "         6.44 

44 

99.98      .28 

3.0-2 

99.77      4.76 

99.58      6.50 

46 

.34 

99.90      3.08 

4.8.' 

99.57      6.56 

48 

.40 

3.14 

99.76      4.88 

99.56      6.61 

50 

"          .45 

3.20 

4.94 

6.67 

52 

"          .51 

99.89      3.26 

99.75      4.99 

09.55      6.73 

54 

.57 

3.31 

99.74      5.05 

99.54      6.78 

56 

99.97       .63 

"         3.37 

5.11 

99.53      6.84 

58 

.09 

99.88      3.43 

99.73      5.17 

99.52      6.90 

60 

.74 

3.49 

5.23 

99.51      6.96 

c+/=    .75 

.75       .01 

.75        .02 

.75        .03 

.75        .05 

c-f/=  1.00 

1.00      .01 

1.00        -.03 

1.00        .04 

1.00        .06 

c+/=  1.25 

1.25       .0> 

1.25         -03 

1.25        .05 

1.25        .08 

TABLE    XXI.    STADIA    REDUCTIONS 


107 


TABLE  XXI. 

STADIA  REDUCTIONS    FOR   READING    100 


Minutes 

4° 

5° 

6° 

7° 

Her.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

0' 
2 

4 
6 
8 
10 

99.51      6.96 
7.02 
99.50      7.0? 
99.49      7.13 
99..4S        .19 
99.47        .25 

99.24      8.68 
99.23      8  74 
99.22      8.80 
99.21      8.85 
99.20      8.91 
99.19      8.97 

98.91     10.40 
98.90    10.45 
68.88    10.51 
98.87    10.57 
9S.86    10  62 
98.85    10.63 

98.5!     12.10 
93.50    12.15 
98.48    12  21 
S8.47    12.20 
98.46     12.3:.' 
98.44    12.38  ! 

12 
14 
16 

18 
•20 

99.46        .-30 
.36 
99.45        .42 
99.44        .48 
99.43        .53 

99.18      9.03 
99.17      9.08 
99.16      9.14 
99.15      9.20 
99.14      9.25 

98.33    10.74 
98.82    10.79 
98.81     10.85 
98.80    10.91 
98.78    10.96 

98.43    12.43 
93.41     12.49 
98.40     12.55 
98.39    12.60 
98.3?     12.66 

oo 
24 

2f> 
28 
30 

99.42        .59 
90.41         .65 
99.40        .71 
99.39        .76 
99.38        .82 

99.13      9.31 
99.11      9.37 
,99.10      9.43 
;99.09      9.48 
99.08      9.54 

98.77    11.02 
98.76     11.08 
98.74     11.13 
98.73    11.19 
98.72    l;.25 

98.36    12.72 
98.34    12.7? 
93.33    12.83 
98.31     12.88 
93.89    12.94 

32 
34 
36 
38 
40 

99.38        .88 
99.3?         .94 
99.36      7.99 
99.35      8.05 
99.34      8.11 

99.07      9.60 
99.06      9.G5 
99.05      9.71 
99.04      9.7? 
99.03      9.83 

98.71     11.30 
9869    11.  30 
98.68    11.  -52 
98.67    11.47 
98.65    11.53 

98.28    13.00 
98.27    13.05 
98.25    13.11 
98.24     13.1? 
98.22    13.22 

42 

44 
46 
48 
50 

99.33      8.17 
99.32      8.22 
99.31      S  28 
99.30      8.34 
99.29      8.40 

99.01      -9.88 
99.00      9.94 
9S.99    10.00 
98.98    10.05 
98.9?    10.11 

98.64    11.59 
98.63    11.64 
98.61     11.70 
98.60    11.76 
98.58    11.81 

98.20    13t28 
98.19     13.33 
93.17    13  39 
98.16    13.45 
98.14    13.50 

52 
54 

56 
58 
60 

99.28      8.45 
99.27      8.51 
99.26      8.57 
99.25      8.63 
9924      8.68 

98.90    10.17 
98.94    10.22 
98.93    10.  2S 
98.92    10.34 
98.91     10.40 

98.57    11.87 
93  .56    11.93 
98:54    11.98 
98.53    12.04 
98.51    12.10 

98.13    13.56 
!>8.11     13.61 
1^.10    13.6? 
9S.08     13.73 
Sfc'.OG    13.78 

c+/=     -T5 
c+/=  1.00 
c  +  /=  1.25 

.75        .06 
I.  00        .OS 
1.25        .10 

.75        .07 
.99        .09 
1.24        .11 

.75        .08 
.99        .11 
1.24        .14 

.74        .10 
.99        .13 
I.:54        .16 

108 


TABLE    XXI.    STADIA    REDUCTIONS 


TABLE  XXI. 

STADIA   KFDUCTIONS   FOR  READING   100 


Minutes. 

8° 

9° 

10° 

11° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

98.06    13.78 
98.05    13.84 
98.03    13,89 
98.01     13.95 
98.00    14.01 
97.98    14.06 

97.55    15.45 
97.53    15.51 
97.52    15.56 
97.50    15.62 
97.48.   15.67 
97.46    15.73 

96.98    17.10 
96.96    17.16 
96.94    17.21 
96.92    17.26 
96.90    17.32 
96.88    17.37 

96.36    18.73 
9634    18.78 
96.32    18.84 
96.29    18.89 
96.27    18.95 
96.25    .19.00 

12 
14" 
16 
18 
20 

97.97    14.12 
97.95    14.17 
97.93    14.23 
97.92    14.28 
97.90    14.34 

97.44    15.78 
97.43    15.84 
97.41     15.89 
97.39    15.95 
97.37    16.00 

96.88    17.43 
96.84    17.48 
96.82    17.54 
96.80  .17.59 
96.78    17.65 

96,23    19.05 
96.21    J9.ll 
96.18    19.16 
90.16    19.21 
96.14    19.27 

22 
24 

26 
28 
30 

97.88    14.40 
97.87    14.45 
97.85    14.51 
97.  a3    14.56 
97.82    14.62 

97.35    16.06 
97.33    16.11 
97.31     16.17 
97.29    16.22 
97.28    16.28 

96.76    17.70 
96.74    17.76 
96.72    17.81 
96.70    17.86 
96.68.  17.92 

96.12    19.82 
96.09    19.38 
96.07    19.43 
90.05    19.48 
96.03    19.54 

32 
34 
36 

38 
40 

97.80    14/67 
97.78    14.73 
97.76    14.79 
97.75    14.84 
97.73    14.90 

97.26    16.33 
97.24    16.39 
97.22    16.44 
97.20    16.50 
97.18    16.55 

96.66    17.97 
96.64    18.03 
96.62    18.08 
96.60    18.14 
96.57    18.19 

96.00    19.59 
95.98    19.64 
95.96    1&.70 
95.93    19.75 
95.91    19.80 

42 

44 
46 
48 
50 

97.71    14.95 
97.69    15.01 
97.68    15.06 
97.66    15.12 
97.64    15.17 

97.16    16.61 
97.14    16.66 
97.12    16.72 
97.10-  16.77 
97.08    16.83 

96.55    18.24 
96.53    18.30 
96.51     18.35 
96.49    18.41 
96.47    18.46 

95.89    19.86 
95.86    19.9! 
95.84    19.96 
95.82    20.02 
95.79    20.07 

52 

i          54 

56 

58 
60 

97.62    15.23 
97.61     15.28 
97.59    15.34 
97.57    15.40 
97.55    15.45 

97.06    16.88 
97.04     36.94 
97.02    16.99 
97.00    17.05 
96.98    17.10 

96.45    18.51 
96.42    18.57 
96.40    18.62 
96.38    18.68 
96.36    18.73 

95.77    20.12 
95.75    20.18 
95.72    20.23 
95.70    20.28 
95.68    20.34 

(.+/=     .75 
t;4-/  =  'i.OO 
c  +  /=  1.25 

.74        .11 
.09        .15 

i.as      .13 

.74        .12 
.99        .16 
1.23        .21 

.74        .14 
.98        .18 
1.23        .23 

.73        .15 
.93        .20 
1.22        .25 

M'jfc, 


TABLE    XXI.     STADIA    REDUCTIONS 


109 


TABLE  XXI. 

STADIA    11EDUCTIONS    FUIi   HEADING    100 


1 

Minutes. 

12° 

13: 

14° 

15° 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

95.68    20.34 
95.65    20.39 
95.63    20.44 
95.61    20.50 
95-58    20.55 
95.56    20.60 

94.94    21.92 
94.91     21.97 
94.89    22.02 
94.86    22.08 
94.84    22.13 
M.81    22.18 

94.15    23.47 
94.12    23.52 
94.09    23.58 
94.07    23.63 
94.04    23.68 
94.01    23.73 

93.30    25.00 
93.27    25.05 
93.24    25.10 
93.21     25.15 
93.18    25.20 
93.16    25.25 

12 
14' 
16 
18 
20 

95.53    20.66 
C5.51    20.71 
95.  49    20.76 
95.46    20.81 
95.44    20.87 

94.79    22.23 
94.76    2-2.28 
94.73    22.34 
94.71    22.39 
94.68    22.44 

93  93    23.78 
93.  55    23  .S3 
93.93    23.88 
93.90    23.93 
93.87    23.99 

93.13    25.30 
93.10    25.35 
93.07    25.40 
93.04    25.45 
93.01    25.50 

22 
24 
26.    . 

2? 
30 

95.41     20.92 
95.39    20.97 
95.36    21.03 
95.34    21.08 
95.32    21.13 

94.66    22.49 
94.63    22.54 
94.60    22.60 
34.58    22.65 
94.55    22.70 

93.84    24.04 
93.81     24.09 
93.79    24.14 
93.76    24.19 
93.73    24.24 

92.98-  25.55 
92.95    25.60 
92.9-2    25.65 
92.89    25.70 
92.86    25.75 

32 
34 
36 
38 
40 

95.29    21.18 
95.27    21.24 
95.24    21.29 
95.22    21  31 
95.19    21.39 

94.52    22.75 
94.50    22.80 
94.47    22.85 
94.44    22.91 
94.42    22.96 

53.70    24.29 
93.67    24.34 
93.65    24.39 
93.62    24.44 
93.59    24.49 

92.83    25.80 
92.80    25  85 
92.77    25.90 
92.74    25.95 
92.71    26.00 

42 
44 

46 
48 
50 

95.17    21.45 
95.14    21.50 
95.12    21.55 
95.09    21.60 
95.07    21.66 

94.39    2301 
94.36    23.06 
94.34    23.11 
94.31    23.16 
94.28    23.22 

93.56    24.55 
93.53    24.60 
93.50    24.65 
93.47    24.70 
93.45    24'.  75 

92  68    26.05 
92.65    26.10 
92.62    26.15 
92.59    26.20 
92.56    26.25 

52 
54 
56 
58 
60 

95.04    21.71 
95,02    2L76 
94.99    21.81 
94.97    21.87 
94.94    21.92 

'94.26    23.27 
94.23    23.32 
94.20    23  37 
94.17    23.42 
94.15    23.47 

93.42    24.80 
93.39    24.85 
93.36    24.90 
93.33    24.95 
93.30    25.00 

92.53    26.30 
92.49    26.35 
92.46    26.40 
92.43    26.45 
92.40    26.50 

c+f=    .75 
c-t-/=  1.00 
c-(-/=1.25 

.73        .16 
.98        .22 
1.22        .27 

.73        .17 
.97        .23 
1.21        .29 

.73        .19 
.97        .25 
1.21         .31 

.72        .20 
•      .96        .27 
1.20        .34 

no 


TABLE    XXI.    STADIA    REDUCTIONS 


TABLE  XXI. 


STADIA   REDUCTIONS  FOR  READING   100 


Minutes. 

13° 

17  ' 

18° 

19° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

H-.r.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

92.40    26.50 
92.37    26.55 
92.34    26.59 
P-"'.31     26.64 
9^.28    26.69 
92.25    26.74 

91.45    27.96 
91.42    28.01 
91.39    28.06 
91.35    28.10 
91.32    28.15 
91.29    28.20 

90.45    29.39 
90.42    29.44 
90.38    29.48 
90.35    29.53 
90.31     29.58 
90.28    29.62 

89.40    30.78 
89  36    30.83 
89.33    30.87 
89.29    30.92 
89.26    30.97 
89.22    3K01 

12 
14 

16 
18 
20 

92.22    26.79 
92.19    2684 
92.15    26.89 
92.12    26.94 
92.09    26.99 

91.26    28.25 
91.22    28.30 
91.19    28.34 
91.16    28.39 
91.12    28.44 

90.24    29.67 
90.21    29.72 
90.18    29.76 
90.14    29.81 
90.11    29.86 

89.18    31.06 
89.15    31.10 
89.11     31.15 
89.08    31.19 
89.04    31.24 

22 
24 
26 

28 
30 

92.06    27.04 
92.03    27.09 
92.00    27.13 
91.97    27.18 
91.93    27.23 

91.09    28.49 
91.06    28.54 
91.  OJ    28.58 
90.99    28.63 
90.96    28.68 

90.07  .29.90 
90.04    29.95 
90.00    30.00 
89'.  97    30.04 
89.93    30.09 

89.00    31.28 
88.96    31.33 
88.93    31.38 
88.89    31.42 
88.86    31.47 

32 
34 
36 
38 
40 

91.90    27.28 
91.87    27.33 
91.84    27.  3S 
91.81    27.43 
91.77    27.48 

90.92    28.73 
90.89    28.77 
90.56    28.82 
90.82    28.87 
90.79    28.92 

89.90    80.14 
89.86    30.19 
,S9  83    30.23 
89.79    30.28 
89.76    3tf.32 

88.82    31.51 
88.78    31.56 
&S.75    31.60 
88.71     31.65 
8^67    31.69 

42 
44 
46 

48 
50 

91.74    27.52 
91.71     27.57 
91.68    27.02 
91.65    27.67 
91.61    27.72 

90.76    28.96 
90.72    29.01 
90.69    29.06 
90.66    29.11 
90.62    29.15 

89.72    30.37 
89.69    30.41 
89.65    30.46 
89.61     30.51 
89.58    30.55 

88.64    31.74 
88.60    31.78 
88.56    31.83 
88.53    81.87 
88.49    31.92 

52 
54 

56 
58 
60 

91.58    27.77 
91.55    27.81 
91.52    27.86 
91.48    27.91 
91.45    27.96 

90.59    29.20 
90.  M    29.25 
90.52    29.30 
90.48    29.34 
90.45    29.39 

89.54    30.60 
89.51    30.65 
69.47    30.69 
89.44    30.74 
89.40    30.78 

88.45    31.96 
88.41    32.01 
88.38    32.05 
88.34    32.09 
88.30    32.14 

c-f/=     -75 
c-f  /=  I  -00 
c+/  =  1  25 

.72        .21 
.J6        .28 
1.20        .30 

.72        .23 
.95        .30 
1.19        .38 

.71         .24 
.95        .32 
1.1'J        .40 

.71         .25 
.94        .33 

1.18        .42 

TABLE    XXI.    STADIA    REDUCTIONS 


111 


TABLE  XXI. 


STADIA  REDUCTIONS  FOR  READING   100 


Minutes. 

20° 

21° 

22° 

23° 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Eler. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 
2 

4 
6 
8 
10 

88.30    32.14 
88.26    32.18 
88.23    32.;>3 
88.19    32.27 
88.15    32.32 
88.11    32.  3'ti 

87.16    3:5.46 
87.12    33.50 
87.08    33.54 
87.04    3:5.59 
87.00    33.63 
86.96    33.67 

85.97    34.73 
85.93    31.77 
85.89    34.82 
85.85    34.86 
85.80    34  90 
85.76    34.94 

84.73    35.97 
84.69    36.01 
84.65    36.05 
84  01     36.09 
84.57    36.13 
84.52    36.17 

12 
14 
16 
18 
20 

88.08    32.41 
88.04    32.45 
S8.00    32.4!) 
87.56    32.54 
87.93    32.58 

86.92    33  72 
86.88    33  76 
86.84    3.<  80 
86.80    33.84 
86.77    33.89 

85.72    34.98 
85.68    35.02 
85.64    35.07 
85.60    35.11 
85.56    35.15 

84.48    3<?.21 
S4.44    36.  i* 
84.40    3<i.29 
84.35    36.33 
84.31     36.37 

22 
24 
20 

88 

30 

87.89    32.63 
87.85    32.67 
87.81     3\>.?2 
87.77    32.76 
87.74    32.80 

86.73    33  93 
86.69    33.97 
86.65    34.01 
86.61     34.06 
86.57    34.10 

85.52    35.19 
85.48    3.-).  23 
85.44    35.27 
85.40    35.31 
85.36    35  36 

84.27    36.41 
f>4.23    36.45 
84.18    36.49 
84.14    3b.53 
84.10    36.57 

32 
34 
36 
38 

40 

87.70    32.85 
87.66    32.89 
87.62    32.93 
87.58    35.98 
87.54    33.02 

86.53    34.14 
86:49    34.18 
8645    34.23 
86.41     31.27 
86.37    34.31 

85.31    35.40 
85.27    35.44 
85.23    35.48 
85.19    35.52 
85.15    35.56 

84.06    36.C.1 
84.01     36  U5 
83.97    3t>  69 
83.93    36.73 
83.89    36  77 

42    . 
44 

46 
48 
50 

87.51    33.07 
87.47    33.11 
87.43    33.15 
87.39    33.20 
87.35    33.24 

86.33    34.35 
86.29    34.40 
86.25    34.44 
86  21     34  48 
80.17    34.  J2 

85.11     35.60 
85.07    35.64 
85.0-.'    35.6* 
84.98    35.72 
84.94    35.76 

83.84    36.80 
83.80    36.84 
83.76    36.  S8 
83.72    36.92 
83.67    36.96 

52 
54 
56 
58 
60 

87.31    33.28 
87.27    33.33 
87.24    33.37 
87.20    33.41 
87.16    33.46 

86.13    34  57 
86.09    34.01 
86.05    34.6.-) 
80.01     34.  C9 
85.97    34.73 

84.90    35.80 
84.86    35.85 
84.82    35.89 
84.77    35.93 
84.73    35.97 

83.  G3    37.00 
83.59    37  01 
83.51    37  08 
83.50    37.1.' 
83.46    37.16 

c+/=    .75 
c-f/=1.00 
c+/=l-25 

.70        .26 
.94        .35 
1.17        .44 

.70        .27 
.93        .37 
1.16        .46 

.69        .29 
.92        .38 
1.15        .48 

.69.       .30 
.92        .40 
1.13         .50 

TABLE    XXI.     STADIA    REDUCTIONS 


TA^BLE   XXI. 


STADIA  SEDUCTIONS    FOTC   HEADING   100 


Minutes. 

24° 

25° 

2CT 

27° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist,     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.  '    Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

83.46    37.16 
83.41    37.20 
83.37    37.23 
83.33    37.27 
83.528    37.31 
83.24    37.35 

82.14    38.30 
82.09    38.34 
82.05    38.38 
82.01    38.41 
81.96    38.45 
81.92    38.49 

80.78    39.40 
80.74    39.44 
80.69    39.47 
80.65    39.51 
80.60    39.54 
80.55    39.58 

79.39    40.  «' 
79.34    40  49 
79.30    40.52 
79.25    40.55 
79.20    40.59 
79.15    40.62 

.     12 
14 
16 
18 
20 

83.20    37.39 
83.15    37.43 
83.11    37.47 
83.07    37.51 
83.02    37.54 

81.87    38.53 
81.83    38.56 
81.78    38  60 
81.74    38.64 
81.69    38.67 

80.51    39.61 
80.46    39.65 
80.41     39.69 
80.37    39.72 
80.32    39.76 

79.1.1     40.66 
79.06    40.  69 
79.01     40=72 
78.96    40.76 
78.92    40.79 

22 
24 
26 
28 
30 

82.98    37.58 
82.93    37.62 
82.89    37.66 
82.85    37.70 
82.80    37.74 

81.65    38.71 
81.60    38.75 
81.56    38.78 
81.51-  38.82 
81.47    38.86 

80.28    39.79 
80.23    39.83 
80.18    39.86 
80.14    39.90 
80,09    39.93 

78.87    40.82 
78.82    40.86 
78.77    40  89 
78  73    40.92 
78'.68    40.96 

32 
34 
36 

38 
40 

82.76    37.77 
82.72    37.81 
82.67    37.85 
82.63    37.89 
82.58    37  ..93 

81,42    38.89 
81.38    38.93 
81.33    38.97 
81.28    39.00 
81.24    39.04 

.80.04    39.97 
80.00    40.00 
79.95    40.04 
79.90    40.07 
79.86    40.11 

78.63    40.99 
78.58    41.02 
78.54    41.06 
78.49    41.09 
78.44    41.12 

42 
44 
46 
48 
50 

82.54    37.96 
82.49    38.00 
82.45    38.04 
82.41    38.08 
82.36    38.11 

81.19    39.08 
81.15    39.11 
81.10    39.15 
81.06    39.18 
81.01    39.22 

79.81    40.14 
79.76    40.18 
79.72    40.21 
79.67    40.24 
79.6-2    40.28 

78.39    41.16 
78.34    41.19 
78.30    41.22 
78.25    41.26 
78.'20    41.29 

52 
54 
56 
58 
60 

82.32    38.15 
82.27    38.19 
82.23    38.23 
82.18    38.26 
82.14    38.30 

80.97    39.26 
80.92    39.29 
80.87    39.33 
80.83    39.36 
80.78    39.40 

79.58    40.31 
79.53    40.35 
79.48    40.38 
79.44    40.4^ 
79.39    40/45 

78.15    41.3-2 
78.10    41.35 
78.06    41.39 
78.01     41.42 
77.96    41.45 

.c-f/=    -75 
c+/  =  1.00 
c+/=1.25 

.68        .31 
.91        .41 
1.14        .52 

.68        .32 
.90        .43 
1.13        .54 

.67        .33 
.89        .45 
1,13        -Kfl 

.66        .35 
.89        .46 
1.11        .03 

A    MANUAL   OF    LAND   SURVEYING. 


T  =  R  tan.  H 
T  _  50  tan.  ^ 
Sin.  D 

Sin.  D  =  — 
K 

Sin  D 

Cl 

JRVE    FORMUL, 

R  =  T  cot.  M  I 
R  — 

E. 

Chord  def.-cbord9a 
R 

No.  chords  -  — 
D 

Tan.  def.=  &  chord  def. 

Sin.  D 
jt:x>q 

E  =  R  ex.  sec.  H  I 
E  —  T  tan.  H  I 

T 

The  square  of  any  distance,  divided  by  twice  the  ra- 
dius, will  equal  the  distance  from  Tangent  to  Curve, 
very  nearly. 

Table  XX  contains  Tangents  and  Externals  to  a  1° 
curve.  Tan.  and  Ext.  to  any  other  radius  may  be  found, 
nearly  enough,  by  dividing  the  Tan.  or  Ext.  opposite 
the  given  Central  Angle  by  the  given  degree  of  curve. 

To  find  Deg.  of  Curve,  having  the  central  Angle  and 
Tangent :  Divide  Tan.  opposite  the  given  Central  An- 
gle by  the  given  Tangent. 

To  find  Deg.  of  Curve,  having  the  Central  Angle  and 
External :  Divide  Ext.  opposite  the  given  Central  An- 
gle by  the  given  External. 

To  find  Nat.  Tan.  and  Nat.  Ex.  Sec.  for  any  angle  by 
Table  XX  :  Tan.,  or  Ext.  of  twice  the  given  angle  di- 
vided by  the  radius  of  a  1°  curve  will  be  the  Nat.  Tan. 
or  Ex.  Sec. 

To  find  angle  for  a  given  distance  and  deflection. 

Rule  1.  Multiply  given  distance  by  .01745  (def.  for  1° 
for  1  ft.),  and  divide  given  deflection  by  the  product. 

Rule  2.  Multiply  given  deflection  by  57.3,  and  divide 
the  product  by  the  given  distance. 

To  find  deflection  for  a  given  angle  and  distance: 
Multiply  the  angle  by  .01745.  and  the  product  by  the 
distance. 


MANUAL 

—  OF  — 

LAND  SURVEYING 


BY  F.  HODGMAN,  M.  S.,  C.  E., 

Practical  Surveyor  and  Civil  Engineer. 

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